A POWER SAVING FOR POLYNOMIAL DIFFERENCES AT PRIME ARGUMENTS
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For a prime-intersective polynomial h of degree k >= 2, sets of
integers up to N with no nonzero difference of the form h(p) for p
prime have size at most C_h N^{1-c_h}. The proof combines number
theory (primes in arithmetic progressions), harmonic analysis
(reflection-positive tuple laws and Fourier lifts), and a descent
on shorter intervals. Earlier work gave subpolylogarithmic savings;
this account provides a full proof of a fixed positive saving.
Contents
0. Statement and outline
1. Auxiliary polynomials and local admissibility
2. Reflection-positive laws on residue tuples
3. Fourier lifts and multilinear estimates
4. Prime estimates
5. Kernel comparison and pair cancellation
6. Descent and the proof of the main theorem
Conventions. h is an integer polynomial, degree k >= 2, leading
coefficient a > 0. We write e(x) = exp(2 pi i x). For xi in Q/Z we
write q(xi) for its reduced positive denominator, with q(0) = 1.
Norms and expectations on finite sets use uniform probability unless
another measure is specified. Implicit constants and lower thresholds
may depend only on h (or on structural parameters fixed from h) and
are uniform in the auxiliary index l and the residue data unless
noted. N^{o(1)} denotes a quantity bounded by C_eps N^eps for every
eps > 0, with this same uniformity.
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0. Statement and outline
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Definition 0.1. An integer polynomial h is prime-intersective if for
every q >= 1 there is an integer r with (r, q) = 1 and
h(r) = 0 mod q.
Theorem 0.2 (main). Let h be prime-intersective of degree k >= 2 and
positive leading coefficient. Then there are constants c_h > 0 and
C_h >= 1, depending only on h, such that for every N >= 1 and every
A subset of {1, ..., N},
(A - A) intersect { h(p) : p prime } subset of {0}
implies |A| <= C_h * N^{1 - c_h}.
The convention at zero allows a prime root of h: if h(p) = 0 for some
prime p, the value 0 imposes no further restriction because every
A - A already contains 0.
Examples.
- h(x) = (x - 1)^2 is prime-intersective: r = 1 works for every q.
Its prime values h(p) = (p - 1)^2 are nonzero squares for every prime
p. A set with no nonzero square differences automatically avoids
them, so Theorem 0.2 gives a power saving for square-difference-free
subsets of [N].
- h(x) = x^2 + 1 is not prime-intersective at q = 3: for r coprime
to 3, h(r) is congruent to 2 mod 3, never 0. The theorem's
hypothesis fails, so no bound is claimed for this h.
History. For the intersective (integer-argument) case there is a rich
literature (Furstenberg 1977, Sarkozy 1978). The prime-argument case,
which is weaker (fewer forbidden differences), already has substantial
history (Sarkozy on p - 1, Lucier, Ruzsa and Sanders, Wang, Green on
p + 1, Thorner and Zaman, Rice on all prime-intersective polynomials
of degree k >= 2 with subpolylog savings). What we add is a fixed
power saving c_h > 0.
Outline of the proof. The argument proceeds in five stages. The
first four build the analytic and combinatorial objects; the fifth
combines them into the main theorem.
(i) Local (Section 1). Choose a p-adic unit root z_p of h. Rescale
and shift to h_l. Define "admissible residues" at each prime power,
and build a zero-sum cycle of length L independent of l and p.
(ii) Tuples (Section 2). Construct a reflection-positive
probability measure mu_{l,p} on finitely many residue labels.
Designated cycle increments are admissible; an exceptional
submeasure eps_{l,p} is captured; conditional mixing holds.
(iii) Lifts (Section 3). Lift the interval indicator 1_A to
residue spaces via bounded-denominator Fourier sums. Reflection
positivity gives a signed Holder inequality and a seminorm. The
mixing estimate gives large-denominator decay. Uniform lift
moments control all needed norms.
(iv) Primes and kernel (Sections 4-5). Use a zero-free half-plane
for L-functions to control primes in arithmetic progressions, and
Vaughan plus Weyl to control primes over polynomial phases. These
build a signed kernel on prime-supported values of h_l that
matches a model kernel of admissible differences to accuracy
H^{-gamma/2}.
(v) Descent (Section 6). Combine avoidance and the tuple-integral
quantity; obtain a recurrence that contracts on shorter intervals.
Uniform induction gives the power saving.
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1. Auxiliary polynomials and local admissibility
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The first obstruction is that restricting a set A to a residue
class changes the polynomial that expresses its forbidden
differences. We choose progression steps (powers of lambda) so that
the transformed polynomials stay in a common auxiliary family. We
then identify "admissible" local values at every prime power, which
will supply the designated cycle of the tuple laws in Section 2.
1.1. The p-adic root and the rescaled polynomial.
By prime-intersectivity, for each prime p the equation
h(x) = 0 mod p^j has a solution x coprime to p for every j >= 1.
These solutions are compatible under reduction, so by a
compactness (finitely branching tree) argument, there is a p-adic
unit z_p in Z_p^* with h(z_p) = 0.
Expand
h(z_p + X) = sum_{j = m_p}^k b_{j, p} X^j, b_{m_p, p} != 0,
with 1 <= m_p <= k and coefficients b_{j, p} in Z_p.
Define a completely multiplicative function lambda : N -> N by
lambda(p) = p^{m_p}.
For l >= 1 define r_l in (-l, 0] by the Chinese remainder theorem:
r_l = z_p mod p^{v_p(l)} for every p | l,
where v_p(l) is the p-adic valuation of l. Then (r_l, l) = 1. For
l = 1 set r_1 = 0.
Put
T_l(x) = r_l + l * x,
h_l(x) = h(T_l(x)) / lambda(l),
a_l = a * l^k / lambda(l).
Lemma 1.1.
(a) h_l is in Z[x], of degree k, with leading coefficient
a_l > 0 and all coefficients O_h(a_l).
(b) h_1 = h and T_1(x) = x.
(c) For every D >= 1 there is i in {0, ..., D-1} with
T_l(D x - i) = T_{l D}(x),
h_l(D x - i) = lambda(D) * h_{l D}(x),
D | lambda(D), lambda(D) <= D^k.
(d) b_{m_p, p} is a unit in Z_p for every p >= P_unit (threshold
depending only on h).
Proof. (a) Fix p. If p | l, set u = v_p(l) and make the affine
change of variable
Y = (r_l - z_p) / p^u + (l / p^u) x,
whose coefficients are in Z_p and whose slope is a unit. Direct
substitution into the expansion of h at z_p gives
h_l(x) = [lambda(l) / p^{u m_p}]^{-1} * sum_j b_{j,p}
p^{u(j - m_p)} Y^j.
Every term is in Z_p because b_{j,p} is in Z_p. If p does not divide l, lambda(l)
is a p-adic unit and integrality is automatic. Thus h_l is integral
at every prime, hence h_l in Z[x]. Its leading coefficient is
a * l^k / lambda(l) = a_l > 0. Expanding h(r_l + l x) with
|r_l| < l bounds each coefficient by O_h(l^k / lambda(l))
= O_h(a_l).
(b) r_1 = 0 so T_1(x) = x and h_1(x) = h(x).
(c) r_{l D} = r_l mod l (both reduce to z_p mod p^{v_p(l)} for
p | l, and are zero mod p for p does not divide l). Both lie in (-l D, 0], so
r_{l D} = r_l - i l for some integer 0 <= i < D. Substituting:
T_{l D}(x) = r_{l D} + l D x = r_l + l(D x - i) = T_l(D x - i).
Also h_{l D}(x) = h(T_{l D}(x)) / lambda(l D), hence
lambda(D) * h_{l D}(x) = h(T_{l D}(x)) / lambda(l)
= h_l(D x - i).
The bounds D | lambda(D) and lambda(D) <= D^k come from
1 <= m_p <= k.
(d) Factor h = c * prod_i g_i^{d_i} over Z[x] with c != 0 an integer
and g_i distinct primitive irreducible nonconstant polynomials.
Exclude the finitely many primes dividing c, or dividing one of the
nonzero integer right sides of cleared Bezout identities for g_i, g'_i
and for g_i, g_j (i != j). For any remaining prime p, the root z_p
sits at exactly one factor g_i: g_i(z_p) = 0, g'_i(z_p) is a unit,
and every other g_j(z_p) is a unit. Then m_p = d_i and the
coefficient of X^{m_p} in the expansion of h(z_p + X) equals
c * g'_i(z_p)^{d_i} * prod_{j != i} g_j(z_p)^{d_j} * (unit from
multinomial expansion), which is a p-adic unit. QED
1.2. Forbidden set and avoidance in progressions.
Define the forbidden set at parameter l:
D_l = { h_l(m) : m in Z, m >= 1, T_l(m) prime }.
We call A subset of [N] avoiding at l if
(A - A) intersect D_l subset of {0}.
The hypothesis of the main theorem corresponds to avoidance at
l = 1: since h_1 = h and T_1(m) = m, the set D_1 is exactly
{ h(p) : p prime, p >= 1 }.
The point of the auxiliary family is that avoidance survives
restriction to well-chosen progressions.
Lemma 1.2 (Transport to progressions). For any l, D, n >= 1 and any
integer c, the set
A' = { x in [n] : lambda(D) x + c in A }
is avoiding at l D whenever A is avoiding at l.
Proof. Suppose to the contrary that A' is not avoiding at l D.
Then there are x_1, x_2 in A' and m' >= 1 with T_{l D}(m') prime,
such that x_1 - x_2 = h_{l D}(m'), which is nonzero. Set
a_i = lambda(D) x_i + c, so both are in A. Lemma 1.1(c) gives
a_1 - a_2 = lambda(D) (x_1 - x_2) = lambda(D) h_{l D}(m')
= h_l(D m' - i)
for a unique 0 <= i < D. Also D m' - i >= D - (D - 1) = 1 and
T_l(D m' - i) = T_{l D}(m') is prime, so h_l(D m' - i) is in D_l.
It is a nonzero difference of two elements of A (since lambda(D) > 0
and h_{l D}(m') != 0). This contradicts avoidance of A at l. QED
1.3. Admissible values modulo a prime power.
Fix once and for all
delta = 1 / (100 * k * 2^k), gamma = delta / 10,
constants depending only on h.
Call x in F_p "allowed regular" if p does not divide T_l(x) and F'(x) != 0 mod p,
where F = h_l mod p.
Lemma 1.3 (Local exponential sums). There is P_1 > max(k, 2),
depending only on h, such that for every l >= 1 and every prime
p >= P_1, at least p/2 arguments in F_p are allowed regular, and
for every integer v coprime to p:
|E_{x mod p^j, p does not divide T_l(x)} e(v h_l(x) / p^j)|
<= p^{-8 delta j} (j >= 1),
|E_{x mod p, x allowed regular} e(v h_l(x) / p)|
<= p^{-4 delta}.
Proof. Take the threshold large enough that h_l mod p is
nonconstant of positive degree with h_l' mod p not identically zero.
Nonconstancy is uniform in l: if p does not divide l, it follows from p does not divide a and the
definition; if p | l, it reduces to a unit multiple of Y^{m_p}. The
derivative excludes at most k - 1 residues, so at most k arguments
fail to be allowed regular; this gives the p/2 count.
Modulo p. Let k' be the degree of F = h_l mod p. The complete mean
S = E_{x mod p} e(v F(x) / p) vanishes if k' = 1. If k' >= 2, Weyl
differencing gives
|S|^{2^{k'-1}} <= (k' - 1) / p.
Indeed, square and group by the increment u = x' - x a total of
k' - 1 times. The final inner sum over x is 1 unless
v * k'! * (leading coeff of F) * u_1 * ... * u_{k'-1}
is a multiple of p, i.e. unless some u_i is zero; the probability
of that is at most (k'-1)/p.
Restricting to allowed-regular arguments changes the mean by at most
k/p. So for p >= P_1 (with P_1 also making k/p <= p^{-4 delta}
absolutely small):
|E_{allowed} e(v F(x)/p)|
<= (k'-1)^{2^{-(k'-1)}} p^{-2^{-(k'-1)}} + O_k(p^{-1})
<= p^{-4 delta},
thanks to 8 delta < 2^{-(k-1)} which holds for
delta = 1/(100 k 2^k).
For j >= 2. Put n = floor(j/2) and write x = x_0 + p^{j - n} y with
x_0 mod p^{j - n} and y mod p^n. Taylor expansion over the integers:
F(x) = F(x_0) + p^{j - n} y F'(x_0) mod p^j.
The condition p does not divide T_l(x) depends only on x_0. Averaging over y kills
every term with p^n does not divide F'(x_0). Only x_0 with p^n | F'(x_0) survive.
Count surviving x_0 by interpolation. Put b = ceil(n / (k - 1)). The
solutions of F'(x_0) = 0 mod p^n occupy at most k - 1 residue classes
modulo p^b. For otherwise, choose k distinct classes with solutions
x_1, ..., x_k, and apply Lagrange interpolation to F' (degree
<= k - 1):
F'(x) = sum_i F'(x_i) * prod_{s != i} (x - x_s) / (x_i - x_s).
The p-adic valuation of each denominator prod_{s != i} (x_i - x_s)
is at most (k - 1)(b - 1) < n, and each numerator F'(x_i) is divisible
by p^n; so every coefficient of F' is divisible by p, contradicting
F' not identically zero mod p.
Since b <= n <= j - n, this class count applies to the full range of
x_0 modulo p^{j - n}. The unit condition retains at least half of the
arguments, so the surviving proportion is at most
2 (k - 1) p^{-b} <= 2 (k - 1) p^{-j / (3 (k - 1))}.
Because 8 delta < 1/(3(k-1)) (from our choice of delta), one more
increase of P_1 gives <= p^{-8 delta j}. QED
Lemma 1.4 (Uniform lifting classes). For every prime p < P_1 there
is an integer S_p >= 0 (depending only on h and p) such that for
every l >= 1 some integer x satisfies
p does not divide T_l(x) and v_p(h_l'(x)) <= S_p.
Set
E_p = 2 S_p + 1 if p < P_1, E_p = 1 if p >= P_1.
Let p be any prime and let x satisfy the displayed condition for
p < P_1, or be allowed regular for p >= P_1; put s = v_p(h_l'(x)).
Then:
(i) Every integer y in the class x mod p^{E_p - s} satisfies the
unit condition p does not divide T_l(y), and its h_l-value is congruent to
h_l(x) mod p^{E_p}.
(ii) For every j > E_p, the uniform distribution on this class mod
p^j pushes forward under h_l to the uniform distribution on
{ y' mod p^j : y' = h_l(x) mod p^{E_p} }.
Proof. Small primes. Fix p < P_1. For a nonconstant polynomial
over Z_p, the minimum valuation of its nonconstant coefficients is
preserved by an affine substitution of unit slope (the two
inequalities follow by expanding the substitution and its inverse).
The p-adic form of h_l (from Lemma 1.1(a)) is obtained from h by a
unit affine substitution when p | l, and is trivially integral when
p does not divide l; in either case this minimum is bounded uniformly in l by a
constant depending only on h and p. Differentiating raises it by at
most max_{1 <= j <= k} v_p(j), so the minimum coefficient valuation
of F' has an upper bound C_p independent of l.
Take any k distinct integers x_1, ..., x_k in [1, k p] satisfying
p does not divide T_l(y) (there are at least k). Lagrange interpolation of F'
(degree <= k - 1) through these points has denominators whose p-adic
valuations are bounded by some B_p independent of l. If every
F'(x_r) had valuation greater than C_p + B_p, then every coefficient
of F' would have valuation greater than C_p, contradicting the
definition of C_p. So S_p = C_p + B_p works.
Both p ranges. Taylor expansion gives
F(x + p^{E_p - s} y) = F(x) + p^{E_p} (c_1 y + p C(y))
with c_1 = p^{-s} F'(x) coprime to p, and C(y) in Z[y] because each
term of degree d >= 2 gains a factor p^{E_p - 2s} >= 1. Set
G(y) = c_1 y + p C(y). For integers y != z,
(G(y) - G(z)) / (y - z) = c_1 + p * (C(y) - C(z)) / (y - z)
is coprime to p, so v_p(G(y) - G(z)) = v_p(y - z). Hence G is
bijective on every finite residue ring Z / p^m.
(i) For y in the class x mod p^{E_p - s}, write y = x + p^{E_p - s} z.
The expansion shows F(y) = F(x) mod p^{E_p}. Since E_p - s >= 1, the
class preserves the unit condition: T_l(y) = T_l(x) + l * p^{E_p - s} z
is prime to p because p does not divide T_l(x) and p does not divide l * p^{E_p-s} when the latter
is prime to p.
(ii) For j > E_p, uniform y mod p^{j - E_p + s} corresponds to
uniform z mod p^{j - E_p}; each z mod p^{j - E_p} occurs p^s times.
The permutation G is bijective mod p^{j - E_p}, so pushing forward
gives the uniform distribution on the specified residue class
of h_l(x) mod p^j. QED
Definition. A residue y mod p^{E_p} is admissible at l if y = h_l(x)
mod p^{E_p} for some integer x satisfying the conditions of
Lemma 1.4. For a modulus that is a product of distinct factors
p^{E_p}, a residue is admissible if it is admissible at every factor.
Lemma 1.5 (Admissible zero-sum cycle of length L). Choose an integer
L > 2 satisfying
4 delta L > 2 and p^{E_p} | L for every p < P_1.
Then for every l >= 1 and every prime p there are L admissible
residues y_0, ..., y_{L-1} mod p^{E_p}, with repetitions allowed,
such that
y_0 + ... + y_{L-1} = 0 mod p^{E_p}.
Proof. For p < P_1: By Lemma 1.4, an admissible residue exists.
Choose one, say y_0, and repeat: y_j = y_0 for all j. The sum is
L * y_0 = 0 mod p^{E_p} because p^{E_p} | L.
For p >= P_1: choose L independent uniform allowed-regular arguments
x_0, ..., x_{L-1} in F_p. Then by character orthogonality and
Lemma 1.3 with v = 1,
p * P(F(x_0) + ... + F(x_{L-1}) = 0 mod p) - 1
<= (p - 1) * max_{v != 0} |E_{x_j} e(v sum_j F(x_j) / p)|
<= (p - 1) p^{-4 delta L}.
Since 4 delta L > 2, the right side is < 1 for p >= P_1. Hence the
probability is positive, and some choice of x_j gives the required
sum. QED
Construction of the cycle. Keep for the rest of the paper the fixed
P_1, exponents E_p, integer L, and thresholds S_p. For any larger
prime cutoff P > P_1, the same sequence of length L propagates
admissible residues modulo the product prod_{p < P} p^{E_p} by the
Chinese remainder theorem: at each prime combine the length-L cycle
at that prime; the sum stays 0 modulo the modulus. So a larger
cutoff never requires a new cycle length.
Cycle coordinates. Given y_0, ..., y_{L-1} mod p^{E_p} as in
Lemma 1.5, set
z_0 = 0, z_j = y_0 + ... + y_{j-1} (1 <= j <= L).
Then z_L = z_0. The increment z_{j+1} - z_j = y_j is admissible.
Distinct indices need not carry distinct labels.
The next section builds the reflection-positive tuple laws whose
designated cycle uses exactly these admissible increments.
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2. Reflection-positive laws on residue tuples
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We build a probability measure mu_{l,p} on the space of residue
labels z_v in F_p, v in a large finite set V, with four properties:
reflection positivity, admissible cycle, translation invariance and
covariance, and conditional mixing. The construction is due to the
companion paper on intersective polynomial differences; the change
here is that the cycles must carry admissible values at prime
arguments.
2.1. Slots, cuts, and the statement.
Choose an integer t >= 1 with (t+1) delta / 2 > 5, and set
w = 2 t + 1, K = L!, r_* = K / 2.
Let Pi be the set of permutation words on [L] (a word is a bijection
from the positions [L] to the symbols [L]). The set of slots is
V = Pi^w, d = |V| = K^w,
which is even. The entries of a slot are called blocks. Fix an
L-cycle c on the positions [L]. Composition pi c permutes the
positions of pi. Identifying c^j itself with a word, fix the
designated cycle slots
v_j = (c^j, ..., c^j) for 0 <= j < L, v_L = v_0.
A cut is specified by a block b in [w] and an ordered pair of
distinct symbols alpha, beta in [L]. Its plus side is
V_+ = { v in V : alpha precedes beta in block b },
and theta denotes the involution on V that exchanges alpha and beta
in block b (keeping everything else fixed). Thus
|V_+| = |V_-| = d/2 and theta identifies the two halves.
For a measure nu on F_p^V, the reflection positivity condition at
cut (b, alpha, beta) is: for every real function
J : F_p^{V_+} -> R,
int J((z_v)_{v in V_+}) J((z_{theta v})_{v in V_+}) dnu(z) >= 0.
Proposition 2.1 (Tuple laws). There is a threshold P > P_1,
depending only on h, with the following property. For every l >= 1
and every prime p >= P, there are a probability measure mu_{l,p} on
F_p^V and a submeasure 0 <= eps_{l,p} <= mu_{l,p} such that:
(1) mu_{l,p} is reflection positive. Both mu_{l,p} and eps_{l,p}
are invariant under simultaneous translation z_v -> z_v + a in
F_p. In particular, any nonzero one of the measures
mu_{l,p}, eps_{l,p}, mu_{l,p} - eps_{l,p} has uniform
one-slot marginals after normalization.
(2) Writing eta_{l,p} = eps_{l,p}(1) and
mu_{l,p}^circ = (mu_{l,p} - eps_{l,p}) / (1 - eta_{l,p}),
we have eta_{l,p} <= p^{-4}. Under mu_{l,p}^circ every cycle
increment z_{v_{j+1}} - z_{v_j} (0 <= j < L) is admissible at
l.
(3) For every D >= 1 coprime to p and every a' in F_p, the map
A_{D,a'}(z)_v = lambda(D)^{-1} (z_v - a')
sends mu_{l,p} to mu_{l,D} and eps_{l,p} to eps_{l,D}.
(4) For rho_p in { mu_{l,p}, mu_{l,p}^circ }, every v in V, and
every f : F_p -> C with uniform mean zero,
|| E_{rho_p}[ f(z_v) | (z_u)_{u != v} ] ||_{L^2(rho_p)}
<= p^{-gamma} ||f||_{L^2(F_p)}.
All thresholds and constants are independent of l.
Strategy of proof. Begin with the simplest law: independent uniform
labels, weighted by the values at the designated cycle increments
(to enforce the cycle property) and by nothing else. This law has
the cycle property and conditional mixing (via Fourier bounds from
Lemma 1.3), but is not reflection positive.
To fix reflection positivity, add "forgotten block" components.
When a block is marked (forgotten), the slots in a cut at that block
share a list of labels, which makes the cut matrix diagonal. A
positive interaction between selection indices ensures that the
marked cut quadratic form bounds the mean square of its test
function from below. Comparing with the unmarked cut, whose
contribution can be negative, the marked positive term dominates.
The remaining components (with all t+1 blocks marked) have no cycle
constraints and form the exceptional submeasure.
2.2. Estimates for weighted graphs.
Fix l and p >= P_1. Write F = h_l. Define the edge weight
W(y) = (1/k) # { x in F_p : F(x) = y, x allowed regular }
for y in F_p, and put c_p = E_{y in F_p} W(y).
As F mod p is nonconstant of degree at most k, 0 <= W(y) <= 1 and
c_p >= 1/(2 k). Lemma 1.3 with allowed-regular only gives, using
negative-sign Fourier coefficients on F_p,
|W_hat(a)| <= p^{-4 delta} for every a in F_p \ {0},
because W_hat(a) is proportional to the normalized mean of
e(-a F(x)/p) over the allowed regular arguments.
Lemma 2.2 (Weighted graph estimates). Let G be a fixed finite
directed graph with no loops and with at most one edge on each
unordered pair of vertices. Give its vertices independent uniform
labels Y_x in F_p, and put
I_G(Y) = prod_{x -> y in E(G)} W(Y_y - Y_x), e_G = |E(G)|.
(i) E I_G = c_p^{e_G} + O_G(p^{-4 delta}).
(ii) If the vertices are partitioned into two classes A and B and
every edge joins A to B, regard I_G as a kernel between their
uniform label spaces. Then its associated operator T_D (with
D = I_G - c_p^{e_G}) satisfies
||T_D||_{L^2 -> L^2} <= O_G(p^{-delta}).
Both bounds are uniform in l.
Proof. (i) The convolution kernel W(y - x) - c_p has L^2 operator
norm at most p^{-4 delta} by W_hat(a) <= p^{-4 delta} for a != 0.
Telescope the product I_G by replacing one edge weight at a time by
c_p. In a resulting correction term, hold all labels except those at
the endpoints x, y of the modified edge fixed. Because there is at
most one edge on each pair and no loops, the remaining weight
factorizes as a(Y_x) b(Y_y) with |a|, |b| <= 1. The operator norm
bound on W - c_p controls the conditional expectation by
p^{-4 delta}. Summing over the e_G edges proves (i).
(ii) In uniform L^2 spaces, the fourth Schatten moment of T_D is
tr((T_D T_D^*)^2) = E_{a, a' in A; b, b' in B}
D(a, b) D(a', b) D(a', b') D(a, b').
(The kernel here is D applied to the box A x B.)
All kernels are real-valued. Expand D = I_G - c_p^{e_G} into four
factors. Each resulting summand is (up to a power of -c_p^{e_G})
the expectation E of a product of selected copies of I_G over an
extended vertex set (two independent copies of A and of B). The
formal graph that counts edges is loop-free and has at most one edge
on each unordered pair. So (i) applies to it, giving
E(product) = c_p^{# (edges)} + O_G(p^{-4 delta}).
Replacing all edge weights by c_p makes the entire expanded sum
vanish (the binomial identity for (I - c_p^{e_G})^4 summed over
selectors of copies). Hence the fourth moment is O_G(p^{-4 delta}),
and the largest singular value of T_D (defined below) is at most
the fourth root of this, i.e. O_G(p^{-delta}). QED
2.3. Components by forgetting blocks.
For S subset of [w] with |S| = s <= t + 1, call the blocks in S
marked. For v in V, its address is u_S(v) = tuple of words in the
unmarked blocks; the address set is U_S = Pi^{[w] \ S}.
At every address u take a list
( Y_{u, 1}, ..., Y_{u, ell_s} ), ell_s = r_*^s,
of independent uniform labels in F_p, independently across u. A slot
of address u will choose one of the labels in its list.
For s <= t put a directed edge u -> u' between addresses when
u'_b = u_b c in at least t + 1 unmarked blocks b. (2.1)
Every entry of the first list is joined to every entry of the second
list, with weight
I_S(Y) = prod_{u -> u'} prod_{i, i' in [ell_s]} W(Y_{u', i'} - Y_{u, i}).
For s = t + 1 set I_S = 1. (These are exactly the components with no
cycle constraints.)
There are no loops because c != 1. There are also no opposite edges:
since c^2 != 1, the witnessing blocks for u -> u' and for u' -> u
would be disjoint, requiring 2(t+1) > w blocks, which is impossible.
So the graph on individual list entries satisfies Lemma 2.2.
Next, select indices j_v in [ell_s] uniformly and independently at
all slots. Put Delta = 1/(10 L^2). For distinct slots x, y in V,
set H^S_{xy} = Delta^g if x and y agree in every block except one
marked block, and in that block the two words differ by exchanging
symbols at positions of distance g. In all other cases put
H^S_{xy} = 0, and also H^S_{xx} = 0. Define
R_S(j) = exp( sum_{x < y} H^S_{xy} * 1_{j_x = j_y} ), (2.2)
nu_S(Phi) = E_{Y, j} [ I_S(Y) R_S(j) *
Phi((Y_{u_S(v), j_v})_{v in V}) ], (2.3)
the sum in (2.2) being over unordered pairs of distinct slots.
Each nu_S is a finite nonnegative measure and satisfies
I_S(Y)^{1/2} R_S(j) in [c, C]
for constants c > 0 and C depending only on the fixed slot system,
so nu_S has bounded mass both above and below.
Reflection invariance. Every component is invariant under every
reflection theta. Exchanging two symbols in a block preserves all
transposition distances used in H^S. For an unmarked block the
exchange permutes addresses and preserves (2.1), because symbol
permutations commute with the right action of c on positions. For a
marked block addresses are unchanged. The uniform lists and indices
are preserved in either case. Hence the cut matrices of nu_S are
symmetric. Positivity is the next issue.
2.4. Positive contributions and their comparison.
Fix a cut in block b, and for a real function J of the plus labels
put
Q_S(J) = int J((z_v)_{v in V_+}) J((z_{theta v})_{v in V_+})
dnu_S(z).
Step A (b in S). Suppose b in S, so the cut is at a marked block.
Let A_{xy} = H^S_{x, theta y} for x, y in V_+, so A is the matrix of
interactions across the cut. By reflection invariance A is
symmetric. If the two cut symbols in slot x occupy positions of
distance g in block b, the diagonal contribution is
A_{xx} = Delta^g. Any other transposition taking x across the cut
to some y != x must move one cut symbol past the other while fixing
the other, and can only occur in block b, exchanging positions of
distance at least g+1. There are at most L^2 such transpositions,
so
sum_{y != x} |A_{xy}| <= L^2 Delta^{g+1} = (L^2 Delta) Delta^g
= Delta^g / 10,
using Delta = 1/(10 L^2). Hence A is diagonally dominant:
A >= kappa I in the PSD order, kappa = (9/10) Delta^{L-1} > 0.
(The worst case is g = L-1.)
Now identify minus indices with plus indices using theta, and let
j, j' be two half-index assignments. Put
u(j) = (1_{j_x = i})_{x in V_+, i in [ell_s]}
(the indicator matrix of the half-index assignment). The cross
factor of R_S is the matrix
C(j, j') = exp( u(j)^T (A (x) I) u(j') ).
Split A = kappa I + PSD. The entrywise exponential of a PSD Gram
matrix is PSD: expand its power series and use the Schur product
theorem for each power. The factor coming from A - kappa I is PSD
and has diagonal entries at least 1. The factor from kappa I is the
tensor product, over x in V_+, of
(1)(1)^T + (e^kappa - 1) I,
which dominates (e^kappa - 1)^{d/2} I. Taking the Schur product with
the other (PSD, diagonal >= 1) factor preserves this lower bound,
because Schur multiplication with the identity returns the diagonal
of the other factor. The internal interactions in the two halves
multiply the matrix on the left and right by the same diagonal
matrix with entries at least 1, so the lower bound survives.
Reflected slots have the same address when b in S; for fixed lists
Y the two factors involving J evaluate the same vector, indexed by
half-index assignments (n_s = ell_s^{d/2} of them). The uniform
average over both halves is n_s^{-2} times the associated quadratic
form. Its diagonal lower bound is therefore
(e^kappa - 1)^{d/2} n_s^{-1} times the uniform average of squares.
Since there are only finitely many s <= t + 1, we obtain a fixed
c_* > 0 such that
Q_S(J) >= c_* E_{Y, j_+} I_S(Y) J((Y_{u_S(v), j_v})_{v in V_+})^2
for b in S. (2.4)
Step B (b not in S). Suppose b not in S. The lists split into plus
and minus addresses according to the order of the cut symbols in
block b. Write Y_+ for the plus lists and I_+ = I_+(Y_+) for the
factors of I_S whose edges have both ends on the plus side. Set
A_S(J) = E_{Y_+, j_+} I_+(Y_+) J((Y_{u_S(v), j_v})_{v in V_+})^2.
No index interaction crosses this cut, because changing a marked
block does not change the word in block b. After averaging the
indices, Q_S(J) is the quadratic form of the cross-list edge kernel
applied to the half-function
B(Y_+) = I_+(Y_+) E_{j_+}[R_+(j_+) J((Y_{u_S(v), j_v})_{v in V_+})],
where R_+ is the internal index weight on the plus side.
By Cauchy-Schwarz, R_+ <= C and I_+^2 <= I_+,
||B||_2^2 <= C A_S(J).
Lemma 2.2(ii) replaces the cross kernel by its nonnegative constant
value c_p^{(cross edges)} with operator error O(p^{-delta}). The
constant kernel contributes a nonnegative multiple of (E B)^2.
So, uniformly over S and the cut,
Q_S(J) >= -C p^{-delta} A_S(J) for b not in S. (2.5)
If s = t + 1, the cross kernel is exactly 1, so Q_S(J) >= 0.
Step C (concatenation). Let b not in S with s <= t, and put
S' = S union {b}. We claim that
Q_{S'}(J) >= c_* r_*^{-d/2} A_S(J). (2.6)
The new address forgets block b. Exactly r_* = K/2 old plus
addresses become each new address, one for each word in which alpha
precedes beta. Enumerate those words as sigma_1, ..., sigma_{r_*}
and write the old addresses above a new address u as (u, sigma_a).
Concatenate their old lists by
Ytilde_{u, (a-1) ell_s + i} = Y_{(u, sigma_a), i}
for a in [r_*], i in [ell_s].
These are independent uniform lists of length
r_* ell_s = ell_{s+1}, exactly as required by the definition of nu_{S'}.
Under this identification of list variables, I_{S'} >= I_+
pointwise. Indeed any edge between new addresses is witnessed by at
least t+1 blocks other than b, so every pair of their constituent old
plus addresses already had that directed edge. Thus every factor of
I_{S'} occurs in I_+, and I_+ has additional factors in [0, 1]. If
|S'| = t+1, the same inequality follows from I_{S'} = 1.
In the nonnegative right side of (2.4) applied to S', denote the
independently chosen new indices by j'_v. For a plus slot of old
address (u, sigma_a), restrict j'_v to the interval
{(a-1) ell_s + 1, ..., a ell_s}: this restriction has probability
1/r_*, independently over the d/2 plus slots. Conditional on these
restrictions, the selected labels have exactly their old
distribution. Using I_{S'} >= I_+ gives (2.6). QED
2.5. The mixture.
Define a finite measure and its normalization by
M = sum_{S subset of [w], |S| <= t+1} p^{-|S| delta / 2} nu_S,
mu_{l,p} = M / M(1). (2.7)
Reflection positivity. Fix a cut in block b. Pair each S with
b not in S, |S| <= t, with S' = S union {b}. By (2.5) and (2.6),
the combined quadratic form is at least
p^{-s delta / 2}
( c_* r_*^{-d/2} p^{-delta/2} - C p^{-delta} ) A_S(J) >= 0
for p sufficiently large (delta/2 < delta gives
p^{-delta/2} >> p^{-delta}). Every component containing b is used
in exactly one such pair; the remaining components have |S| = t+1,
b not in S, and nonnegative quadratic forms. So M is reflection
positive, and hence so is mu_{l,p}.
Mass bounds. For S = empty, every list has length one, every
index weight is one, and the component mass is a graph average:
by Lemma 2.2(i) and c_p >= 1/(2k) we have nu_empty(1) >= c_0 > 0
for sufficiently large p, uniformly in l. Bounded masses of the other
components give
0 < c_0 <= M(1) <= C_0.
Define the exceptional submeasure by
eps_{l,p} = ( p^{-(t+1) delta / 2} / M(1) ) sum_{|S| = t+1} nu_S.
It has mass
eps_{l,p}(1) <= C p^{-(t+1) delta / 2} <= p^{-4}
after increasing the prime threshold (by choice of t).
2.6. Cycle property and translation invariance.
For |S| <= t at least t+1 blocks are unmarked, so the addresses of
v_j and v_{j+1} satisfy (2.1) at every one of them. The product I_S
therefore includes W(Y_{u_S(v_{j+1}), j_{v_{j+1}}} - Y_{u_S(v_j), j_{v_j}}).
On the support of W, and in particular on the uniform-in-indices
event j_{v_j} != j_{v_{j+1}}, only y = F(x) for some allowed regular
x contributes, and every such y is admissible mod p by definition.
So under mu_{l,p}^circ every cycle increment z_{v_{j+1}} - z_{v_j}
is admissible at l. This proves (2).
Translation invariance. Translating every list entry by the same
element of F_p leaves all W weights unchanged. Each component, and
hence both measures in Proposition 2.1, is translation-invariant.
The one-slot marginals are uniform because translation acts
transitively on F_p.
2.7. Covariance and conditional mixing.
Covariance. If p does not divide D, the auxiliary-family identity (Lemma 1.1(c))
gives
W_{lD, p}(y) = W_{l, p}(lambda(D) y).
Indeed the substitution x -> D x - i is bijective mod p, preserves
T_l(D x - i) = T_{lD}(x), and preserves the nonvanishing of F'
because both D and lambda(D) are units mod p. Consequently changing
every list entry to lambda(D)^{-1}(Y_{u, i} - a') transforms each
component for l into the component for lD, preserving its mass and
all index weights. The mixtures and their exceptional parts therefore
satisfy (3).
Conditional mixing. First consider the normalized leading component
nu_empty / nu_empty(1). At a fixed slot v write x = z_v and
y = (z_u)_{u != v}. Its unnormalized density, relative to
independent uniform labels, factors as
J_1(x, y) J_2(y),
where J_1 is the product of the m edge weights incident with v and
J_2 contains all remaining edges. Put
A(y) = E_x J_1(x, y), B(y) = E_x f(x) J_1(x, y).
Apply Lemma 2.2(ii) to the graph of incident edges with partition
{ {v}, V \ {v} }. Since E f = 0,
||A - c_p^m||_2 <= C p^{-delta}, ||B||_2 <= C p^{-delta} ||f||_2.
The conditional expectation at y is B(y)/A(y) when A(y) > 0. The
marginal density of y is J_2(y) A(y) / nu_empty(1), so the squared
conditional norm is
(1/nu_empty(1)) E_y J_2(y) |B(y)|^2 / A(y), (2.8)
with the ratio set to zero when A = 0. On A >= c_p^m / 2, use
||B||_2^2 and the lower bound on A. The complementary set has
uniform probability O(p^{-2delta}) by the estimate on A, since
c_p^m is bounded below. On that set, weighted Cauchy-Schwarz and
J_1 <= 1 give
|B(y)|^2 / A(y) <= E_x |f(x)|^2 J_1(x, y) <= ||f||_2^2.
Together with J_2 <= 1 and the lower bound on nu_empty(1), this
shows (2.8) is O(p^{-2 delta}) ||f||_2^2.
Mixtures. Both mu_{l,p} and mu_{l,p}^circ are mixtures of the
normalized components, and in either mixture the total probability
of S != empty is O(p^{-delta/2}). To justify transfer of the
estimate, retain the component label as an additional random
variable. Conditional Jensen shows that conditioning on this label
as well as on all other slots can only increase the squared
conditional norm. In each normalized component the trivial bound is
||f||_2^2 (since that component has a uniform one-slot marginal).
Averaging the leading estimate and these trivial bounds gives
|| E_{rho_p}[ f(z_v) | (z_u)_{u != v} ] ||_2^2
<= C p^{-delta/2} ||f||_2^2.
Since 2 gamma = delta/5 < delta/2 (recall gamma = delta/10), one
more increase of the threshold gives the claimed bound
<= p^{-2 gamma} ||f||_2^2. This proves (4) and completes the proof
of Proposition 2.1. QED
2.8. Constants fixed here.
Fix henceforth a threshold P > P_1 for which Proposition 2.1 holds,
large enough also that
p^{gamma/2} >= 3 (p >= P), prod_{p >= P} (1 + p^{-3}) - 1 <= 1/4,
(t+1) delta / 2 > 5,
where products are over primes. (The tail product goes to 1 because
sum_p p^{-3} converges.) Let
M_0 = prod_{p < P} p^{E_p} >= 2.
A residue mod D = M_0 q_B, with q_B a product of distinct primes at
least P, is admissible at l if it is admissible at every local
factor p^{E_p}. L, E_p, P, P_1, delta, gamma, M_0, t, w, K, r_*
are all fixed from now on.
--------------------------------------------------------------------
3. Fourier lifts and multilinear estimates
--------------------------------------------------------------------
The tuple laws give two complementary ways to control multilinear
averages. Reflection positivity compares arbitrary inputs with a
single diagonal quantity, producing a signed Holder inequality and
a seminorm. Conditional mixing suppresses large Fourier
denominators. We develop both, then apply them to bounded functions
on integer intervals via a bounded-denominator Fourier lift. The
signed-form and lift arguments are the same as in the companion
paper; we reproduce the proofs here for completeness.
3.1. Signed Holder inequality and the seminorm.
Fix l >= 1 and a finite set P of primes at least P. Set
X_P = prod_{p in P} F_p, mu_{l,P} = tensor product of mu_{l,p}.
Collect the prime labels in slot v into a single variable z_v in
X_P, so mu_{l,P} is a probability law on X_P^V. Each one-slot
marginal is uniform. For real functions on X_P write
Z_{l,P}((g_v)_{v in V})
= int prod_{v in V} g_v(z_v) dmu_{l,P},
Z_{l,P}(g) = Z_{l,P}((g)_{v in V}).
The empty prime set is allowed; then X_P is a singleton and
Z_{l,empty}(g) = g^d.
Fourier notation on X_P. Every frequency has a unique additive
decomposition
xi = sum_{p in P} a_p / p mod 1, 0 <= a_p < p,
with character e(z xi) = prod_{p in P} e(a_p z_p / p). The exact
support of xi is {p : a_p != 0}; its reduced denominator is
q(xi) = prod of that support. For B subset of P, q_B =
prod_{p in B} p, q_empty = 1.
Proposition 3.1 (Signed Holder). For every l >= 1, finite set P of
primes at least P, and real functions g, g_v : X_P -> R,
Z_{l,P}(g) >= 0, (3.1)
|Z_{l,P}((g_v)_{v in V})| <= prod_{v in V} Z_{l,P}(g_v)^{1/d}.
(3.2)
Proof. Across any reflection cut of V, the matrix of the product
law mu_{l,P} is the tensor product of prime-law cut matrices, and
each prime-law matrix is PSD by reflection positivity. So the
product cut matrix is PSD.
Fix a cut V = V_+ union V_- with reflection theta, and let
Phi_+, Phi_- be the two fold maps V -> V_+ that fix the chosen half
and reflect the other half. Given an assignment a : V -> {g's}
(functions from a fixed finite list), write Lambda(a) for its
integral. The two real functions on the plus-side labels entering
the cut bilinear form are
F(x) = prod_{v in V_+} a(v)(x_v),
G(x) = prod_{v in V_+} a(theta v)(x_v).
The cut bilinear form evaluates at F, G with values
Lambda(a), Lambda(a circ Phi_+), Lambda(a circ Phi_-). Cauchy-
Schwarz for the PSD matrix gives
|Lambda(a)|^2 <= Lambda(a circ Phi_+) Lambda(a circ Phi_-),
(3.3)
and Lambda(a circ Phi_+) >= 0 and Lambda(a circ Phi_-) >= 0. Setting a constant proves (3.1).
Next we build a finite sequence of fold maps sending every vertex
to a single one. In one block write p_i for the position of symbol
i in a permutation word. For i = 1, ..., L - 1, fold onto the side
where i precedes i + 1; this sorts (p_i, p_{i+1}) leaving all other
entries fixed. Perform the sweep i = 1, ..., L - 1 a total of L - 1
times. A sweep moves the largest entry to the end; once the largest
entries occupy their final positions, the next sweep fixes them and
places the largest remaining entry immediately before them. So
every position array becomes (1, 2, ..., L). Performing these
sweeps in each block gives fold maps Phi_1, ..., Phi_m (in
chronological order) with
Phi_m o ... o Phi_1 (v) = v_*
for every v in V, where v_* has the identity word in every block.
Now fix a finite nonempty list of real functions. Let M be the
maximum of |integral| over all assignments from that list (with
repetition allowed). If M = 0 we are done. Otherwise choose a
maximizing assignment. Each folded integral is nonnegative and at
most M, while their product is at least M^2 by (3.3). So both
factors equal M, in particular either fold preserves maximality.
Iterating the pullbacks in reverse order gives a constant assignment
with integral M. This proves that the maximum absolute mixed
integral equals the largest diagonal value in the list.
For epsilon > 0 apply this conclusion to the list
g_v / (Z_{l,P}(g_v) + epsilon)^{1/d}, v in V.
Each diagonal value in this list is at most one. Multilinearity
gives
|Z_{l,P}((g_v)_v)|
<= prod_v (Z_{l,P}(g_v) + epsilon)^{1/d}.
Let epsilon -> 0. QED
Define
||g||_{l,P} = Z_{l,P}(g)^{1/d}
for real g on X_P. For B subset of P, let E_B g denote uniform
averaging over the coordinates in B, with all other coordinates
fixed. The result may be regarded as a function on X_P or on
X_{P \ B}.
Lemma 3.2 (Seminorm and changes of coordinates). The map
g -> ||g||_{l,P} is a seminorm on real functions on X_P. It
satisfies:
(i) |E g| <= ||g||_{l,P},
(ii) ||E_B g||_{l, P \ B} = ||E_B g||_{l, P} <= ||g||_{l, P}.
(iii) Translations of the input are isometries.
(iv) If D >= 1 is coprime to every prime in P, then for every
a' in X_P,
|| g(lambda(D)^{-1}(. - a')) ||_{l,P}
= || g ||_{l D, P}. (3.4)
Proof. (i) is nonnegativity plus one-slot uniform marginals. For
homogeneity (d even): Z(g+h) expanded by slots and bounded termwise
by (3.2) gives
Z(g + h) <= sum_{S subset V} ||g||^{|S|} ||h||^{d - |S|}
= (||g|| + ||h||)^d.
Taking d-th roots gives the triangle inequality. (ii) Let G_B be the
additive subgroup of X_P consisting of vectors supported on B. Then
(E_B g)(z) = (1/|G_B|) sum_{a in G_B} g(z + a).
The triangle inequality gives the contraction. E_B g is
independent of the B-coordinates, so integrating out the prime laws
in B leaves the same value of Z (and of the seminorm).
(iii) The tuple laws are invariant under simultaneous translation,
so this is immediate. (iv) The covariance (2.1(3)) pushes the
product law at l exactly to that at lD, and lambda(D) is a unit at
each prime. QED
3.2. Large-denominator decay.
Conditional mixing gives a different estimate, valid also after
removing the exceptional part of each prime law. The proof needs
orthogonality between different exact supports; a triangle inequality
over all Fourier coefficients would lose this decay.
Lemma 3.3 (Large-denominator estimate). Fix l >= 1 and a finite
prime set P as before. At each prime choose rho_p = mu_{l,p} or
mu_{l,p}^circ. Let g_v : X_P -> C be arbitrary functions. If, in one
distinguished slot v, every Fourier coefficient at some denominator
<= R vanishes (with R >= 1), then
|int prod_{u in V} g_u(z_u) d(tensor_p rho_p)|
<= R^{-gamma} ||g_v||_2 prod_{u != v} ||g_u||_{2(d-1)}. (3.5)
Proof. At one prime, let m_p be the marginal law of the labels in
all slots other than v, and put
T_p f = E_{rho_p}[ f(z_v) | (z_u)_{u != v} ].
This is a linear map from uniform L^2(F_p) to L^2(m_p). It preserves
constants and sends mean-zero inputs to m_p-mean-zero outputs.
By the mixing estimate (2.1(4)), it has norm <= p^{-gamma} on the
mean-zero Hilbert space.
Tensor these across primes: T = tensor_p T_p. That the tensor
equals the conditional operator for the product law is first clear
for products of one-prime functions (by independence) and then for
all functions by linearity. On the exact-support-B Fourier space,
T acts as q_B^{-gamma} at worst. Further, the images of distinct
exact-support spaces are orthogonal in L^2(tensor_p m_p): at a prime
in the symmetric difference of the two supports, one image factor
is constant and the other has mean zero, and tensoring preserves
orthogonality.
Decompose g_v = sum_{q_B > R} g_{v,B} by exact supports. Domain
summands and images are orthogonal, so
||T g_v||_2^2 = sum_{q_B > R} ||T g_{v,B}||_2^2
<= sum_{q_B > R} q_B^{-2gamma} ||g_{v,B}||_2^2
<= R^{-2 gamma} ||g_v||_2^2.
Condition the integral on the other slots and apply Cauchy-Schwarz.
The remaining factor is bounded by
||prod_{u != v} g_u(z_u)||_2
<= prod_{u != v} (E |g_u(z_u)|^{2(d-1)})^{1/(2(d-1))}
= prod_{u != v} ||g_u||_{2(d-1)},
using Holder and the uniform one-slot marginals. This proves (3.5).
When P is empty, the frequency hypothesis forces g_v = 0 and the
result is immediate. QED
3.3. Fourier lifts.
Let I be a nonempty consecutive integer interval and f : I -> C.
Put
hat f_I(xi) = (1/|I|) sum_{x in I} f(x) e(-x xi).
For B subset of P define the exact-support contribution
f_B(z) = sum_{q(xi) = q_B} hat f_I(xi) e(z xi).
This depends only on the B-coordinates and has mean zero in each of
them. For Q >= 1 the Fourier lift is
L_{I,P,Q} f(z)
= sum_{B subset P, q_B <= Q} f_B(z)
= sum_{xi on X_P, q(xi) <= Q} hat f_I(xi) e(z xi).
Each f_B depends only on the coordinates in B and has zero mean in
every one of them. When f is real, so are all these. Retaining a
support B always means retaining all its frequencies.
Proposition 3.4 (Uniform lift moments). Let P be a finite set of
primes, Q >= 1, and I a consecutive integer interval with |I| >=
10 Q^6. For every integer r >= 1, epsilon > 0, function f : I -> C,
and subfamily F subset of {B subset P : q_B <= Q},
|| sum_{B in F} f_B ||_{2r} <= C_{r, epsilon} ||f||_infty Q^epsilon.
(3.6)
The constant is independent of the prime set, the interval and its
location, the function, and the subfamily.
Proof outline. The zero function is trivial; rescale so ||f||_infty
<= 1. Write N_I = |I| and let u = max{ |B| : q_B <= Q }, so
u is finite because the empty support is allowed.
Step 1 (conditional energy). For J subset of P with q_J <= Q and
any fixed z_J, claim
sum_{B in F, B superset J} E(|f_B|^2 | z_J) <= 4^{|J|}. (3.7)
The conditional expectation averages uniformly over the complement
of J. If B superset J, the coefficient of an outside character eta
with exact support B \ J after fixing z_J equals
c_J(eta; z_J) = (1/N_I) sum_{x in I} f(x) e(-x eta) *
prod_{p in J} (p 1_{x = z_p mod p} - 1).
(The last factor comes from summing the nonzero frequencies at p.)
The character sets attached to different B superset J are disjoint,
so conditional Parseval writes the left side as the squared
Euclidean norm of this coefficient vector.
Expand the product over J' subset of J. Up to sign, the term J'
contributes a vector with coordinates
(q_{J'} / N_I) sum_{x in I, x = z_{J'} mod q_{J'}} f(x) e(-x eta).
Write the progression as x = a + q_{J'} n, with n in a consecutive
interval of length N'. Then N' >= 0.9 N_I / q_{J'} and
N' q_{J'} / N_I <= 1.1. After the substitution the frequencies
q_{J'} eta are still distinct (their denominators are coprime to
q_J), so they number at most Q^2 and are at least Q^{-2} apart on
the circle. The Gram matrix of these characters in normalized L^2
on the progression has diagonal 1 and off-diagonal entries of
modulus at most Q^2 / (2 N') = Q^5 / (1.8 N_I) <= 1/(18 Q). So its
operator norm is at most 1 + 1/(18 Q).
The synthesis map sending (b_eta)_eta to sum_eta b_eta
e(n q_{J'} eta) on the progression (with normalized counting
measure) has Gram matrix equal to the estimate just derived. Its
adjoint is the normalized Fourier analysis map. Using
||f||_infty <= 1 and N' q_{J'} / N_I <= 1.1, the coefficient
vector has Euclidean norm bounded by an absolute constant. The
triangle inequality over 2^{|J|} subsets J' and one final squaring
gives (3.7).
Step 2 (square-function moments). Set
S(z) = (sum_{B in F} |f_B(z)|^2)^{1/2}.
Expand E S^{2r} and fix the supports of the first r - 1 factors;
their union W' has size at most (r - 1) u and their product depends
only on z_{W'}. For the final support B, f_B ignores coordinates
outside B, so integration outside W' picks out exactly
E(|f_B|^2 | z_J) with J = B intersect W'. Every intersection that
occurs satisfies q_J <= q_B <= Q. For each fixed J the sum over B
with B intersect W' = J is bounded by the larger sum over
B superset J in (3.7). There are at most 2^{(r-1)u} possible
intersections. So
E S^{2r} <= C * 2^{(r-1)u} * 4^u * E S^{2r-2}.
The same with r = 1 uses J = W' = empty. Iteration gives
||S||_{2r} <= C_r^{1+u}. (3.8)
Step 3 (recovering the sum). First handle transversal supports.
Fix a support size j >= 1 and color every prime coordinate with one
of j colors. Call B transversal when |B| = j and B contains exactly
one prime of each color. Let T be the set of transversal supports in
F. Take independent uniform pairs (X_p^0, X_p^1) at every prime and
let swap_p interchange the two copies at prime p. Put
D_B = prod_{p in B} (1 - swap_p) f_B(X^0).
Averaging over second copies leaves f_B(X^0), since every other
term in the expansion has mean zero in some copied coordinate.
Jensen gives
||sum_{B in T} f_B||_{2r} <= ||sum_{B in T} D_B||_{L^{2r}(X^0, X^1)}.
Independent swaps preserve the joint law of the copies and multiply
D_B by a product of signs. So for independent uniform signs
epsilon_p,
||sum_{B in T} D_B||_{L^{2r}(X^0, X^1)}
= ||sum_{B in T} D_B prod_{p in B} epsilon_p||
_{L^{2r}(X^0, X^1, epsilon)}.
Scalar sign estimate. For any coefficients (a_i) and any r >= 1,
||sum_i a_i epsilon_i||_{2r}
<= C_r (sum_i |a_i|^2)^{1/2}.
Expand the even moment; surviving terms have each |a_i|^2 with
even multiplicity. For nonnegative coefficients the sum is at most
(2r)!/(2^r r!) (sum_i a_i^2)^r since (2m)! >= 2^m m!. The complex
case follows from (sum a_i epsilon_i)^r (sum bar a_i epsilon_i)^r by
triangle inequality. Iterating over the j color classes, with
Minkowski in the intermediate L^{2r} spaces, gives (for fixed
X^0, X^1)
||sum_{B in T} D_B prod_{p in B} epsilon_p||_{2r}
<= C_r^j (sum_{B in T} |D_B|^2)^{1/2}.
Now expand the swaps in D_B: 2^j terms indexed by a choice of copy
at each color. For any fixed such choice (consistent across all
supports), the selected prime coordinates have the original
uniform product law. Triangle inequality in L^{2r}(l^2(T)) gives
|| (sum_{B in T} |D_B|^2)^{1/2} ||_{2r}
<= 2^j || (sum_{B in T} |f_B|^2)^{1/2} ||_{2r}
<= 2^j ||S||_{2r}.
So the transversal sum has norm at most (2 C_r)^j ||S||_{2r}.
Now average over independent uniform colorings. A fixed support of
size j is transversal with probability j!/j^j >= e^{-j} (since
log j! >= int_1^j log x dx). So the full size-j sum is exactly the
coloring-average of the transversal sum divided by this probability.
Minkowski yields
||sum_{B in F, |B| = j} f_B||_{2r}
<= (2 e C_r)^j ||S||_{2r}.
Including the empty-support term and summing j <= u, (3.8) and
trimming the constants give
||sum_{B in F} f_B||_{2r} <= (C'_r)^{1+u}.
The product of u distinct primes is at least (u+1)!, so
(u+1)! <= Q and therefore u = o(log Q) uniformly in P (as Q -> infty).
For every fixed r and epsilon we have (C'_r)^{1+u}
<= C_{r, epsilon} Q^epsilon. Restoring ||f||_infty proves (3.6). QED
Lemma 3.5 (Specialization of residue coordinates). Let B subset of P,
fix z_B in X_B, and let G : X_P -> C. For 1 <= s < infty,
||G(z_B, .)||_{L^s(X_{P \ B})} <= q_B^{1/s} ||G||_{L^s(X_P)}. (3.9)
In particular, a Fourier lift or a subfamily of its complete supports
satisfies (3.6) after specialization with an extra factor
q_B^{1/(2r)}. The same bound holds after retaining either the
denominators at most a given H >= 1 or those greater than H.
Proof. The fiber z = z_B has uniform probability 1/q_B in X_P. Its
contribution to E |G|^s is q_B^{-1} E |G(z_B, .)|^s; dropping the
other {q_B - 1} nonnegative contributions gives (3.9). For the
subfamily cut, a character with original support C retains exact
support C \ B after specialization, so the two cuts in the
remaining denominators arise from families
{ C in F : q_{C \ B} <= H }, { C in F : q_{C \ B} > H }.
Apply Proposition 3.4 to these and then use (3.9). Combining
coefficients can cancel frequencies but cannot create any outside
the indicated ranges. QED
In particular, specializing the same set B of coordinates in every
slot (with possibly different values) and applying (3.5) to the
product law on the remaining primes incurs total specialization
factor
q_B^{1/2} * (q_B^{1/(2(d-1))})^{d-1} = q_B.
Two specialized L^2 norms incur the same factor. All moment orders
needed below are fixed once h is fixed; finite products of their
subpower bounds stay subpower, and none depends on l, on the
specialized values, or on the retained supports.
--------------------------------------------------------------------
4. Prime estimates
--------------------------------------------------------------------
Two uniform estimates go into the kernel construction (Section 5).
First, an asymptotic formula for primes in residue classes. Second,
cancellation in polynomial exponential sums over primes.
Both depend on a zero-free half-plane for Dirichlet L-functions.
Let Lambda' be the weighted prime indicator: Lambda'(n) = log n if n
is prime, else 0. Let Lambda be the usual von Mangoldt function:
Lambda(n) = log p if n = p^j for some prime p and j >= 1, else 0.
All sums below run over integers; sums with real endpoints are
interpreted via summation by parts. Constants in this section do not
depend on the lower coefficients of polynomial phases.
4.1. The zero-free input.
Theorem 4.0 (Zero-free input; from the companion paper). For every
integer b >= 1 and every Dirichlet character chi mod b, the Dirichlet
L-function L(s, chi) has no zero in Re s > 7/8. This includes
imprimitive characters and the trivial character of modulus 1 (whose
L-function is zeta(s)). A pole at s = 1 for a principal character is
allowed.
We only use its assertion about L-functions; see the accompanying
paper for its proof.
4.2. Uniform distribution in residue classes.
Proposition 4.1. Uniformly for Y >= 2, 0 <= x <= Y, integers b >= 1,
and reduced residue classes c mod b,
sum_{n <= x, n = c mod b} Lambda'(n)
= x / phi(b) + O(Y^{31/32} log^2(2 b Y)). (4.1)
For a nonreduced class modulo b > 1, the sum is O(log b) and has
zero main term.
Proof. Set sigma_0 = 15/16. We first prove the logarithmic
derivative estimate
|(L'/L)(s, chi)| << log(2 b (2 + |Im s|)) (4.2)
on Re s = sigma_0, and throughout sigma_0 <= Re s <= 2 when
|Im s| >= 1. Constants are absolute, including for imprimitive
characters.
Write delta_chi = 1 for principal chi and 0 otherwise, and
G_chi(s) = (s - 1)^{delta_chi} L(s, chi).
Fix radii 2 - sigma_0 < r < R < 2 - 7/8 and use disks of radius R
centered at s_t = 2 + i t. Theorem 4.0 makes G_chi holomorphic and
nonvanishing on each disk (with the principal pole removed). To bound
its modulus, use periodicity to write
sum_{n <= u} chi(n) = bar chi * u + E_chi(u), |E_chi(u)| <= 2 b,
bar chi = phi(b)/b if delta_chi = 1, else 0.
Partial summation continues the Dirichlet series to Re s > 0:
L(s, chi) = bar chi * s / (s - 1)
+ s int_1^infty E_chi(u) u^{-s-1} du.
On the disk, |G_chi(s)| <= C [2 b (2 + |t|)]^C. At the center, the
Euler product gives |L(2 + i t, chi)| >= zeta(4)/zeta(2), so
|G_chi(s_t)| is bounded below by a positive constant. Choose a
holomorphic logarithm and subtract its value at s_t. Its real part
is at most C log(2 b (2 + |t|)). Borel-Caratheodory on an
intermediate disk, then Cauchy's estimate on the disk of radius r,
gives
|(G_chi'/G_chi)(s)| << log(2 b (2 + |t|)) for |s - s_t| <= r.
Subtracting delta_chi/(s - 1) proves (4.2).
Smooth sum. Because the contour integral must be absolutely
convergent, use smoothing by (1 - u)_+. Mellin inversion gives, for
x >= 1,
sum_{n <= x} (x - n) Lambda(n) chi(n)
= (1/(2 pi i)) int_{(2)} -(L'/L)(s, chi) x^{s+1}/(s(s+1)) ds.
The interchange with the Dirichlet series is absolute. Move the
line to Re s = sigma_0. The horizontal integrals are bounded by
C x^3 log(2 b (2 + T))/T^2, which tends to zero as T -> infty.
The only crossed pole is at s = 1 for a principal character; its
residue is x^2/2. On the new vertical line, integration of
log(2 b (2 + |t|))/(1 + t^2) gives
sum_{n <= x} (x - n) Lambda(n) chi(n)
= delta_chi * x^2 / 2 + O(x^{1 + sigma_0} log(2 b)). (4.3)
Orthogonality. For (c, b) = 1, character orthogonality yields
A_c(x) = sum_{n <= x, n = c mod b} (x - n) Lambda(n)
= x^2 / (2 phi(b)) + O(x^{1 + sigma_0} log(2 b)). (4.4)
(The phi(b) character errors sum and divide by phi(b), giving the
stated uniformity in b.)
Sharp differencing on nonnegative sums. Positivity of Lambda
avoids the delicate complex character problem. For 0 < H < x,
(A_c(x) - A_c(x - H))/H
<= sum_{n <= x, n = c mod b} Lambda(n)
<= (A_c(x + H) - A_c(x))/H.
Take H = x^{31/32}. The main terms differ from x/phi(b) by O(H),
and the errors are O(x^{1 + sigma_0}/H * log(2 b))
= O(x^{31/32} log(2 b)). Bounded x is harmless.
Higher prime powers. Finally,
sum_{p^j <= Y, j >= 2} log p << Y^{1/2} log^2(2 Y).
Removing these from the summable contributions proves (4.1). If a
prime lies in a nonreduced residue class mod b, it divides b; the sum
of logarithms of distinct such primes is at most log b. This gives
the last assertion. QED
4.3. Minor-arc estimate over primes.
Proposition 4.2. Fix k >= 2 and 0 < rho < 1. There is a constant
c_m = c_m(k, rho) in (0, 1/4) with the following property. If Y is
sufficiently large, I subset of [1, Y] is an interval, and f in R[x]
has degree at most k with coefficient b_k of x^k, then
|sum_{n in I} Lambda'(n) e(f(n))| <<_{k, rho} Y^{1 - c_m}, (4.5)
provided theta = (k!)^2 b_k has an approximation a/q satisfying
(a, q) = 1, |theta - a/q| <= q^{-2}, Y^rho <= q <= Y^{k - rho}.
(4.6)
The proof uses Vaughan's identity to split the prime sum into sums
with one short variable and bilinear sums, and Weyl differencing to
turn each into averages of linear phases. We isolate the two
elementary tools first; both are classical.
Lemma 4.3 (Rational spacing). Suppose (a, q) = 1, q >= 2, and
|theta - a/q| <= q^{-2}. For R >= 1 and M >= 1,
sum_{1 <= r <= R} min(M, ||theta r||^{-1})
<< (R/q + 1)(M + q log(2 q)). (4.7)
Proof. Within a block of floor(q/2) consecutive integers, two
distinct indices differ by a nonzero d with |d| < q/2. Then
||theta d|| >= ||a d / q|| - |d|/q^2 >= 1/(2 q),
so the associated points are 1/(2 q) separated on the circle. The
nearest point to zero contributes at most M, and summing others in
bands of width 1/(2 q) gives O(q log(2 q)). Partitioning indices
into such blocks proves (4.7). QED
Divisor counting. When we group a fixed number of nonzero integer
factors by their product, and that product has absolute value at most
Y^{C_k}, the number of such factorizations is Y^{o(1)} (uniformly in
the product): its prime factorization bounds the number by a
fixed-order divisor function, up to a bounded number of sign choices.
Lemma 4.4 (One-variable and mixed differencing). Let k >= 2 and let
f in R[x] have degree at most k with coefficient b_k of x^k.
(i) If J is an interval of integers of length at most an integer
M >= 1, and m is a positive integer,
|(1/M) sum_{n in J} e(f(m n))|^{2^{k-1}} <<_k (1/M^k)
sum_{|u_1|, ..., |u_{k-1}| < M}
min(M, ||k! b_k m^k u_1 ... u_{k-1}||^{-1}). (4.8)
(ii) Let A, B be positive integers, I subset of [1, Y] an interval,
and |a_m|, |d_n| <= 1. Set
S = sum_{A <= m < 2A, B <= n < 2B}
a_m d_n 1_{m n in I} e(f(m n)),
theta = (k!)^2 b_k.
Then
|S/(A B)|^{2^{2k-1}} <<_k (1/(A^k B^k))
sum_{|u_1|, ..., |u_k| < B; |v_1|, ..., |v_{k-1}| < A}
min(A, ||theta u_1 ... u_k v_1 ... v_{k-1}||^{-1}). (4.9)
Zero shifts are included.
Proof. We give (ii); (i) is the same with only one variable.
For a two-variable function g, define
Delta^n_u g(m, n) = g(m, n + u) - g(m, n),
Delta^m_v g(m, n) = g(m + v, n) - g(m, n).
Let D_0 = {(m, n) : m n in I, (m, n) in the box}. Extend all
coefficients by zero outside their boxes.
Cauchy-Schwarz in m:
|S/(A B)|^2 <= (1/A) sum_{A <= m < 2A}
|(1/B) sum_n d_n 1_{D_0}(m,n) e(f(m n))|^2.
Expand the square, group by u = n' - n:
<= (1/B) sum_{|u| < B} |(1/(A B)) sum_{m, n}
d_{n+u} bar d_n 1_{D_0}(m, n + u) 1_{D_0}(m, n)
e(Delta^n_u f(m n))|.
For fixed u the new weight depends only on n and is bounded by one,
so another n-difference is allowed. In particular the product
cutoff mn in I is preserved; it is not replaced by a rectangular
domain.
Induction over n-shifts. For shifts u = (u_1, ..., u_j), let
D_u = {(m, n) : (m, n + eps * u) in D_0 for every eps in {0,1}^j},
F_u(m, n) = Delta^n_{u_j} ... Delta^n_{u_1} f(m n).
The accompanying coefficient d_u(n) is defined recursively:
d_empty(n) = d_n, d_{(u, u')}(n) = d_u(n + u') bar d_u(n).
It is bounded by one. Cauchy-Schwarz in existing shifts (unitary
in L^2 after normalization) and in m, followed by another square
expansion in n, adds one more n-shift. Iteration gives
|S/(A B)|^{2^k} <<_k (1/B^k) sum_{|u_1|, ..., |u_k| < B}
|(1/(A B)) sum_{m, n} d_u(n) 1_{D_u}(m, n) e(F_u(m, n))|.
All summands still have m and n in their original intervals, since
the zero vertex of every translated cube belongs to D_0.
After k differences in n, every lower-degree term vanishes, and
F_u(m, n) = k! b_k u_1 ... u_k m^k.
For 0 <= j <= k-1 and m-shifts v = (v_1, ..., v_j), define
D_{u, v} = {(m, n) : (m + eta * v, n + eps * u) in D_0
for every eta in {0,1}^j, eps in {0,1}^k},
T_{u, v} = (1/(A B)) sum_{m, n} 1_{D_{u, v}}(m, n)
e(Delta^m_{v_j} ... Delta^m_{v_1} F_u(m, n)).
D_{u, empty} = D_u. To remove d_u, Cauchy-Schwarz in n:
|(1/(A B)) sum_{m, n} d_u(n) 1_{D_u}(m,n) e(F_u(m,n))|^2
<= (1/B) sum_{B <= n < 2B}
|(1/A) sum_m 1_{D_u}(m,n) e(F_u(m,n))|^2.
Expanding the square bound yields
A^{-1} sum_{|v_1| < A} |T_{u, (v_1)}|.
This is the first m-differencing step, so removing d_u costs no
extra squaring. For 1 <= j <= k-2, Cauchy-Schwarz in n and the same
square expansion give
|T_{u, v}|^2 <= (1/A) sum_{|v| < A} |T_{u, (v, v)}|.
Applying Cauchy-Schwarz also over existing shifts (whose total
normalized mass is bounded by a constant depending on k), we get
|S/(A B)|^{2^{k+j}} <<_k
(1/(B^k A^j))
sum_{|u| < B, |v| < A} |T_{u, v}|, for 1 <= j <= k-1.
Take j = k-1. For fixed n, the section of D_{u, v} is an interval
of m's: each condition puts m + eta * v in [A, 2A) and its product
with the positive number n + eps * u in I; the intersection has
length at most A. The final phase
Delta^m_{v_{k-1}} ... Delta^m_{v_1} F_u is linear in m with
coefficient
(k!)^2 b_k u_1 ... u_k v_1 ... v_{k-1}.
Geometric summation bounds each m-sum by the displayed minimum
divided by A; summing over at most B values of n, the remaining
normalized sum is that minimum divided by A B. The shift weights
factor as B^{-k} A^{-(k-1)}, proving (4.9). (i) is the same
argument in one variable: k-1 square expansions remove the
auxiliary n-variable, leaving a final linear phase
k! b_k m^k u_1 ... u_{k-1} and normalizer M^{-k}. QED
4.4. Proof of Proposition 4.2.
We first prove estimate (4.5) with Lambda in place of Lambda'.
Put e_0 = rho / (100 k) and Z = Y^{e_0}. For an arithmetic function
g, write g_{<= Z}(n) = g(n) 1_{n <= Z} and g_{> Z} = g - g_{<= Z}.
Let mu be the Mobius function, 1 the constant arithmetic function, *
Dirichlet convolution. Vaughan's identity, in the form used here, is
Lambda = Lambda_{<= Z} + mu_{<= Z} * log
- mu_{<= Z} * Lambda_{<= Z} * 1
+ mu_{> Z} * 1 * Lambda_{> Z}. (4.10)
(From log = 1 * Lambda and mu * 1 = 1_{n=1}.)
Two sums with a short variable. The middle two terms of (4.10) are
sums with outer variable m <= Z^2 and inner variable n <= Y/m. The
outer coefficient is mu(m) or sum_{a b = m; a, b <= Z} mu(a)
Lambda(b), hence of size Y^{o(1)}. The inner weight is log n or 1.
Fix m, set M = ceil(Y/m), and consider any interval subsum
S_m = sum_{n in J} e(f(m n)).
Differencing Lemma 4.4(i). Using
min(M, ||k! b_k r||^{-1}) <= k! min(M, ||theta r||^{-1})
(since ||k! u|| <= k! ||u||) and grouping nonzero-shift tuples by
their product r = m^k u_1 ... u_{k-1} <= m^k M^{k-1}, Lemma 4.3
gives (uniformly in the interval)
|S_m/M|^{2^{k-1}} << Y^{o(1)} ( 1/M + m^k/q + m^k/M
+ 1/M^{k-1} + q/M^k ). (4.11)
Since m <= Y^{2 e_0} and M >= Y^{1 - 2 e_0},
m^k/q, q/M^k <= Y^{-rho + 2 k e_0},
m^k/M <= Y^{-1 + 2(k+1) e_0}.
The remaining terms also save a positive power. Taking the
2^{k-1}-st root gives S_m << M Y^{-c} for some c = c(k, rho) > 0,
uniform in the subinterval. Partial summation handles log n with an
extra O(log Y). Summing over m <= Z^2:
sum_{m <= Z^2} ceil(Y/m) << Y log(2 Y).
Both short-variable terms save a fixed power after including the
outer coefficients.
Bilinear sum. If AB < Y^{1 - e_0} the trivial estimate suffices;
otherwise assume AB >= Y^{1 - e_0}. For the final term of (4.10),
group the factors as (mu_{> Z} * 1)(m) Lambda_{> Z}(n). Each
coefficient is Y^{o(1)} when m n <= Y; a nonzero coefficient forces
m > Z and n > Z. Partition into dyadic boxes
A <= m < 2A, B <= n < 2B, A, B >= Z/2, A B <= Y.
There are O(log^2(2 Y)) relevant boxes. Divide out coefficient
bounds to reduce to |a_m|, |d_n| <= 1, then apply (4.9).
Nonzero-shift tuples group by their absolute product, at most
R = A^{k-1} B^k. By Lemma 4.3 and the divisor bound,
|S/(A B)|^{2^{2k-1}} << Y^{o(1)}
( 1/A + 1/B + 1/q + 1/R + q/(A R) ). (4.12)
(Each nonzero-shift bound before expansion is
Y^{o(1)}/(A R) * (R/q + 1)(A + q log(2 q)); the term
O(log(2 q)/A) is absorbed.) Now A, B >= Y^{e_0}/2, and
A R = (A B)^k >= Y^{k(1 - e_0)},
q/(A R) <= Y^{-rho + k e_0}.
So (4.12) saves a fixed power, uniformly over boxes.
The remaining term Lambda_{<= Z} costs O(Z log(2 Z)). This proves
(4.5) for Lambda with a positive exponent depending only on k, rho.
Removing higher prime powers costs O(Y^{1/2} log^2(2 Y)). Decreasing
the exponent if necessary gives c_m in (0, 1/4) and proves (4.5). QED
--------------------------------------------------------------------
5. Kernel comparison and pair cancellation
--------------------------------------------------------------------
Fix the tuple laws of Section 2. Along each cycle edge v_j ->
v_{j+1}, they prescribe a distribution of admissible increments.
We now realize the low-denominator Fourier multipliers of this
distribution by a signed kernel supported on prime arguments of
the auxiliary polynomial. Avoidance of these prime-supported values
then gives cancellation for pairs of Fourier lifts. The kernel
comparison follows the companion paper's integer-argument method;
the additional ingredient here is the unit conditions at prime
arguments, handled through the unit means and local regular
arguments already developed.
5.1. Parameters.
Fix, depending only on h:
eta = 1/(10000 k),
c_m = c_m(k, eta/4) (from Proposition 4.2),
nu = min(eta, c_m/k) / 10,
beta = nu / 100, zeta = nu / 100,
tau = beta gamma / (1000 k (d+1)),
sigma = gamma tau / 4.
(eta here is distinct from the exceptional masses eta_{l,p}.)
For scale N set
Q = N^beta, H = N^{beta/2},
P_N = { p prime : P <= p <= Q }.
The three exponents tau, sigma, zeta are reserved for the descent.
5.2. The edge multiplier.
Fix l >= 1, write F = h_l, and choose a directed cycle edge
v_j -> v_{j+1}. For each p >= P let pi_p be the F_p-distribution of
z_{v_{j+1}} - z_{v_j} under mu_{l,p}^circ; it is supported on
admissible residues by Proposition 2.1(2). Define, for integer v,
Gamma_p(v/p) = sum_{y in F_p} pi_p(y) e(v y / p).
For v coprime to p (nonzero character), the conditional mixing
estimate (2.1(4)) applied to the second endpoint and paired with the
conjugate character at the first gives
|Gamma_p(v/p)| <= p^{-gamma}.
For rational xi with squarefree denominator supported on primes
>= P, write xi = sum_{p | q(xi)} xi_p (unique mod 1) and put
Gamma(xi) = prod_{p | q(xi)} Gamma_p(xi_p). Then
Gamma(0) = 1, |Gamma(xi)| <= q(xi)^{-gamma}. (5.1)
The multiplier may depend on l and the chosen edge; all estimates
below are uniform in both.
5.3. Pair cancellation.
Proposition 5.1. Suppose N is sufficiently large, l <= N^zeta, and
D = M_0 q_B <= N^beta where q_B = prod_{p in B} p for a finite set
B of primes at least P. Let J_1, J_2 : [N] -> C with |J_i| <= 1,
supported on residue classes a'_1, a'_2 mod D. Assume that
r = a'_2 - a'_1 mod D is admissible at l, and
J_1(x) J_2(y) != 0 implies x < y and y - x not in D_l.
Set
Xi = { xi : q(xi) <= H, q(xi) squarefree, (q(xi), D) = 1 }.
Then for the multiplier Gamma of any directed cycle edge,
D | sum_{xi in Xi} Gamma(xi) hat J_1(-xi) hat J_2(xi) |
<< H^{-gamma/2}. (5.2)
Because M_0 contains every prime < P, the multiplier is defined on
Xi.
5.4. The model kernel.
For the data of Proposition 5.1, set
k_0(n) = (D/N) 1_{1 <= n <= N} 1_{n = r mod D}
sum_{xi in Xi} Gamma(xi) e(-xi n). (5.3)
The congruence restriction in k_0 will be imposed on polynomial
arguments by a local weight; the factors Gamma_p will be imposed by
further weights of unit mean one.
5.5. The prime-supported kernel.
The auxiliary family gives large values F(m) = h_l(m) up to lambda(l)
times the target. Put U = (N/a_l)^{1/k}. Since a_l is a positive
integer and a_l <= a l^k,
U <= N^{1/k}, U >> N^{1/k - zeta}.
By Lemma 1.1(a) there is an absolute x_* such that for x >= x_*,
F(x) ~ a_l x^k, F'(x) ~ k a_l x^{k-1}, F''(x) = O(a_l
x^{k-2}),
and so there is a unique X >= x_* with F(X) = N, with X ~ U. Put
Y = C l U for a fixed constant C chosen so that T_l(X) <= Y. The
relation a_l = a l^k / lambda(l) gives
Y >> N^{1/k}, Y << N^{1/k + zeta}, Y^k ~ N lambda(l). (5.4)
In particular Y <= N for large N.
5.6. Unit means and local weights.
For a periodic function of an integer m, its unit mean mod b
averages over m mod b satisfying (T_l(m), b) = 1. There are exactly
n_b = b prod_{p | b, p does not divide l} (1 - 1/p) = phi(l b)/phi(l)
such classes (a prime p | l excludes no argument because
(r_l, l) = 1; a prime p does not divide l excludes one class). These means factor
over prime powers and are consistent under replacing b by a
multiple.
At a prime p | D: choose an argument witnessing admissibility of
r mod p^{E_p} and put s_p = v_p(F'). Lemma 1.4 supplies a class
mod p^{E_p - s_p} on which unit condition holds and F = r mod
p^{E_p}; uniform argument lifts in this class give uniform higher
lifts of the value. Combining by CRT gives a class mod
d_r = prod_{p | D} p^{E_p - s_p} | D.
Let rho_D be n_{d_r} times its indicator. Then rho_D has unit mean
one and ||rho_D||_infty <= D.
At a prime p does not divide D (so p >= P): for each y with pi_p(y) > 0, choose
one allowed regular argument x_y mod p with F(x_y) = y. Define
psi_p(x_y) = n_p pi_p(y), psi_p(x) = 0 at all other classes.
It has unit mean one, and its F-value distribution is exactly pi_p.
Put
B_p = psi_p - 1, B_u = prod_{p | u} B_p, B_1 = 1,
where u is squarefree and (u, D) = 1. Each B_p has unit mean zero,
B_u has period u, and |B_u| <= u.
Expanding psi_p = 1 + B_p produces a product sum over squarefree u.
In a unit mean against a rational phase e(alpha F(m)), a prime
factor of u absent from q(alpha) makes the corresponding term
vanish. So retaining u <= H keeps every term that could contribute
when the D-free part of q(alpha) is at most H. Denominators greater
than H will be bounded separately.
5.7. The prime kernel. Define
k_1(n) = sum_{x_* < m <= X} 1_{n = F(m)} F'(m)/N *
phi(l)/l * Lambda'(T_l(m)) * rho_D(m) *
sum_{u <= H, squarefree, (u, D) = 1} B_u(m). (5.5)
Every nonzero term has 0 < F(m) <= N and T_l(m) prime, so k_1 is
supported on D_l intersect [N]. No positivity is required; the
kernel is signed.
5.8. Kernel comparison.
Lemma 5.2. For the data of Proposition 5.1,
sup_{alpha in R/Z} | S_0(alpha) - S_1(alpha) | << H^{-gamma/2},
where S_i(alpha) = sum_{n in Z} k_i(n) e(alpha n) are positive-sign
transforms on R/Z.
Proof. Use major arcs of radius 3 N^{-1 + eta} around reduced
rationals with denominator at most N^eta. These are disjoint for
large N (3 eta < 1). The remainder: minor arcs. Bound both
transforms there, then compare coefficients on major arcs.
Minor arc for S_0. Expanding the congruence indicator in (5.3)
gives the exact identity
S_0(alpha) = sum_{xi in Xi} sum_{j' mod D}
Gamma(xi) e(-j' r / D) V_N(alpha - xi + j'/D), (5.6)
where V_N(y) = N^{-1} sum_{n in [N]} e(y n). There are O(D H^2)
terms. Their centers xi - j'/D are distinct: an equality would give
a rational with denominator both coprime to D (as divisor of q(xi))
and dividing D (as a j'/D center). Each center has denominator at
most DH <= N^{3 beta/2} <= N^eta. On minor arcs, the geometric
summation bound for V_N(y) gives
|S_0(alpha)| << D H^2 N^{-eta}. (5.7)
Minor arc for S_1. Consider, with any linear twist y', the sums
sum_{m in I} Lambda'(T_l(m)) e(alpha F(m) + y' m)
over subintervals I of (x_*, X]. On setting n = T_l(m), expand the
restriction n = r_l mod l as the average of its l additive
characters. Each resulting prime sum has polynomial phase in n with
coefficient of n^k equal to b_k = alpha a / lambda(l); the twist
changes only the linear coefficient (because k >= 2).
Fix a representative of alpha, apply Dirichlet approximation to
theta = (k!)^2 b_k with Q_0 = floor(Y^k N^{-eta/2}). The resulting
reduced a'/q satisfies q <= Q_0 and |theta - a'/q| <= q^{-2}. By
(5.4) this places alpha within O(N^{-1 + eta/2}) of a rational with
denominator at most (k!)^2 a q. A minor-arc point must have that
denominator exceeding N^eta for large N, so with Y <= N,
Y^{eta/4} <= q <= Y^{k - eta/4}.
Proposition 4.2 then gives O(Y^{1 - c_m}), uniformly in the linear
twists and interval. Since Y = C l U and (5.4) bounds U below,
Y^{1 - c_m} << U l N^{-c_m/k} <= U N^{zeta - c_m/k} << U N^{-nu}.
Adding the F'/N weight (sup + total variation O(U^{-1})) and the
weights rho_D, B_u by partial summation retains O(U^{-nu}). Their
Fourier l^1 norms: rho_D is constant times one indicator, l^1 <=
n_{d_r} <= D; B_u bound by period times sup, l^1 <= u^2. So
|S_1(alpha)| << D sum_{u <= H} u^2 N^{-nu} << D H^3 N^{-nu}. (5.8)
The powers in (5.7) and (5.8) are 2 beta - eta and 5 beta/2 - nu.
Both are less than -beta gamma / 4 (using beta, zeta <= eta/1000,
eta = 1/(10000 k), gamma < 1), so both bounds are O(H^{-gamma/2}).
Major arcs. Write alpha = alpha_0 + y on a major arc, with
q = q(alpha_0) <= N^eta and |y| <= 3 N^{-1 + eta}. Split
alpha_0 = alpha_D + alpha_1 into the prime-power denominator parts
supported on primes dividing / not dividing D. The coefficient in
(5.6) for the center alpha_0, or zero if that center is absent, is
c_0(alpha_0) = e(alpha_D r) Gamma(alpha_1)
if q(alpha_D) | D and alpha_1 in Xi, and 0 otherwise. (5.9)
Every other center is at circle distance at least 1/(q D H) from
alpha_0, and |y| q D H = o(1), so geometric summation in each of
the O(D H^2) terms gives
S_0(alpha) = c_0(alpha_0) V_N(y) + O(q D^2 H^3 / N). (5.10)
Coefficients for S_1. For squarefree u <= H coprime to D, let A_u
be the unit mean of
b_u(m) = rho_D(m) B_u(m) e(alpha_0 F(m)).
Its period divides b = q D u, and ||b_u||_infty <= D u. For
x_* <= x' < x'' <= X, Proposition 4.1 gives
sum_{x' < m <= x''} b_u(m) (phi(l)/l) Lambda'(T_l(m))
= A_u (x'' - x')
+ O(q D^2 u^2 Y^{31/32} log^2(2 l b Y)). (5.11)
Indeed one class m = c mod b maps to r_l + l c mod lb in the prime
interval of length l(x'' - x'); it is reduced iff (T_l(c), b) = 1
(since (r_l, l) = 1). On reduced classes, phi(l)/l changes the main
term to (x'' - x')/n_b, matching A_u; on nonreduced classes only
finitely many primes divide lb, and their total logarithmic weight
contributes O(log(lb)) with zero main term.
The smooth factor F'(m) e(y F(m))/N has sup plus total variation
O((1 + N|y|)/U). Partial summation in (5.11) again yields main term
(A_u/N) int_{x_*}^X F'(x) e(y F(x)) dx
= (A_u/N) int_{F(x_*)}^N e(y t') dt'.
The normalized integral N^{-1} int_{F(x_*)}^N e(y t') dt' differs
from V_N(y) by O(a_l/N + (1 + N|y|)/N): the missing initial interval
has length F(x_*) << a_l, and comparing integral and integer sum on
[0, N] costs O(1 + N|y|). Hence
S_1(alpha) = c(alpha_0) V_N(y) + O(H^{-gamma/2}),
c(alpha_0) = sum_{u <= H, squarefree, (u,D) = 1} A_u. (5.12)
The error bounds: since Y^{31/32}/U << l N^{-1/(32 k)}, the summed
progression error (over at most b = q D u classes with |weights|
<= D u and u <= H) is
(1 + N|y|)/U * q D^2 sum_{u <= H} u^2 Y^{31/32} N^{o(1)}
<< N^{2 eta + 4 beta + zeta - 1/(32 k) + o(1)}.
The replacement of the integrals is
O(D H^2 (U^{-k} + N^{eta - 1})) << N^{2 beta + k zeta - 1}
+ N^{2 beta + eta - 1}.
The error in (5.10) is at most N^{eta + 7 beta/2 - 1}. All three
are O(N^{-beta gamma / 4}): indeed beta, zeta <= eta/1000,
eta = 1/(10000 k), gamma < 1, and the first exponent has a margin
that absorbs the o(1) loss.
Comparing c and c_0. At primes p | D, the common factor in every A_u
is
chi_D(alpha_0) = e(alpha_D r) if q(alpha_D) | D, else 0. (5.13)
Indeed the p-adic unit-mean base class has prescribed F-value
r mod p^{E_p} and unit mean 1: a character of conductor at most
p^{E_p} therefore returns r, while a primitive character of higher
conductor averages to zero because Lemma 1.4 gives uniform higher
lifts of F on the class, all satisfying the unit condition.
At p does not divide D. Put q_1 = q(alpha_1). If u has a prime factor outside
q_1, the factor B_p has unit mean zero and A_u = 0. For p^j || q_1,
define
a_{p,j} = E_{m mod p^j, p does not divide T_l(m)} e(alpha_p F(m)),
b_{p,j} = E_{m mod p^j, p does not divide T_l(m)} B_p(m) e(alpha_p F(m)).
The surviving A_u (over squarefree u | q_1) are chi_D(alpha_0) times
products that choose b_{p,j} for p | u and a_{p,j} otherwise.
Moreover
a_{p,j} + b_{p,j} = Gamma_p(alpha_p) if j = 1, and 0 if j >= 2.
(5.14)
(For j = 1, psi_p = 1 + B_p pushes unit mean forward to pi_p.
For j >= 2, psi_p is supported on allowed regular arguments modulo
p, and their uniform argument lifts give uniform lifts of the
F-value, so the primitive character has mean zero.)
If q_1 <= H, every squarefree divisor needed in the product
expansion of c already occurs in the sum for c, so c = c_0 (this
also covers nonsquarefree q_1: both sides vanish). If q_1 > H, the
model coefficient c_0 = 0. Bound |c|: Lemma 1.3 gives
|a_{p,j}| <= p^{-8 delta j}, while (5.1) and (5.14) give
|b_{p,j}| <= 2 p^{-gamma j} (for j >= 2, b_{p,j} = -a_{p,j}).
Bounding the truncated product expansion by the full absolute sum:
|c(alpha_0)| <= prod_{p^j || q_1} (|a_{p,j}| + |b_{p,j}|)
<= prod 3 p^{-gamma j} <= q_1^{-gamma/2}
< H^{-gamma/2},
where we used p^{gamma/2} >= 3 for p >= P (Section 2.8). Together
with (5.10) and (5.12) this gives the major-arc comparison. The
minor-arc bounds already proved complete the lemma. QED
5.9. Proof of Proposition 5.1.
Extend J_1, J_2 by zero to Z and put
B_i = (1/N) sum_{x, n in Z} k_i(n) J_1(x) J_2(x + n), i = 0, 1.
Let tilde J_i(alpha) = sum_x J_i(x) e(-alpha x) with counting
measure. Character orthogonality gives
B_i = (1/N) int_{R/Z} S_i(alpha) tilde J_1(-alpha) tilde J_2(alpha)
dalpha.
Both counting-measure l^2 norms are at most sqrt N. Cauchy-Schwarz,
Plancherel, and Lemma 5.2 give |B_1 - B_0| << H^{-gamma/2}. The
forbidden-support hypothesis gives B_1 = 0.
For every contributing pair in B_0, ordering and residue classes
force 1 <= n <= N and n = r mod D. Substituting (5.3) and setting
y = x + n gives
B_0 = (D/N^2) sum_{xi in Xi} Gamma(xi)
sum_{x, y} J_1(x) J_2(y) e(-xi (y - x))
= D sum_{xi in Xi} Gamma(xi) hat J_1(-xi) hat J_2(xi),
which is exactly (5.2). QED
--------------------------------------------------------------------
6. Descent and the proof of the main theorem
--------------------------------------------------------------------
We combine the pair cancellation of Section 5 with the seminorm of
Section 3. The quantity to be decreased is a diagonal tuple
integral of a Fourier lift. Restriction to progressions compares it
with the same quantity at a shorter interval, and avoidance forces
a fixed proportion of residue assignments to cancel (the rest being
paid for by the exceptional submeasure). This is the recurrence
method of the companion paper, adapted here so that the prime
argument survives throughout the auxiliary family (Lemma 1.2).
6.1. The descent quantity.
Keep all structural parameters fixed: M_0, d, t, w, L, delta,
gamma from Section 2, and eta, c_m, nu, beta, zeta, tau, sigma from
Section 5. At every integer scale N >= 1 use P_N, Q = N^beta,
H = N^{beta/2} as in (5.1). For l >= 1 and A subset [N] define
Y_l(N, A) = Z_{l, P_N}( L_{[N], P_N, Q} 1_A ),
Y_l(N) = max over A subset [N] avoiding at l of Y_l(N, A). (6.1)
(The letter Y_l here is unrelated to the range Y of Section 5.)
By the seminorm lemma (Lemma 3.2(i)), the constant Fourier
coefficient of the lift gives
(|A| / N)^d <= Y_l(N, A). (6.2)
For the upper bound, Holder and Proposition 3.4 (uniform lift
moments) give
0 <= Y_l(N) <= N^{o(1)}, for every l >= 1. (6.3)
No restriction of the form l <= N^zeta is imposed in this step.
6.2. Averaging over residues mod M_0.
For each s mod M_0 put
f_s(x) = M_0 1_A(x) 1_{x = s mod M_0},
G_s = L_{[N], P_N, Q} f_s.
Averaging f_s over uniform s gives 1_A. Thus, for independent
uniform residues s_v mod M_0 indexed by v in V, multilinearity gives
Y_l(N, A) = E_{(s_v)} Z_{l, P_N}((G_{s_v})_{v in V}). (6.4)
We shall bound the integrals in (6.4) in two ways. The comparison
with shorter intervals (Prop 6.2) holds for every l; cancellation
(Prop 6.3) needs l <= N^zeta.
6.3. Residue specialization.
Lemma 6.1. Let N >= 1, A subset [N], s mod M_0, I subset of [N] a
nonempty consecutive integer interval. Let B subset of P_N, write
q_B = prod_{p in B} p, D = M_0 q_B, S = P_N \ B. Fix the
B-coordinates of L_{I, P_N, Q} f_s at b = (b_p)_{p in B}, obtaining
g_I : X_S -> R. Let a' mod D be the unique residue satisfying
a' = s mod M_0 and a' = b_p mod p for p in B. Then every frequency
xi on X_S with q(xi) q_B <= Q satisfies
hat g_I(xi) = (D/|I|) sum_{x in I, x = a' mod D} 1_A(x) e(-x xi).
(6.5)
Proof. The coefficient on the left is the sum of
hat f_s_I(xi + omega) e(b omega) over the frequencies omega on X_B
retained by the lift. The condition q(xi) q_B <= Q retains every
omega because q(xi + omega) = q(xi) q(omega) <= Q. Orthogonality of
the q_B characters imposes x = b_p mod p for every p in B and
contributes the factor q_B. Together with the factor M_0 in f_s,
this gives (6.5). QED
6.4. Comparison with shorter intervals.
Proposition 6.2. For all sufficiently large N, every l >= 1, and
every A subset of [N] avoiding at l, the following holds. Let
B subset of P_N satisfy q_B <= N^tau, and put
D = M_0 q_B, L_D = lambda(D),
n_B = ceil(N / L_D), Lambda_B = L_D n_B / N,
S = P_N \ B.
For each v in V choose s_v mod M_0 and fix the B-coordinates of
G_{s_v} arbitrarily, obtaining g_v : X_S -> R. Then
| Z_{l, S}((g_v)_{v in V}) |
<= Lambda_B^d Y_{lD}(n_B) + O(N^{-sigma}), (6.6)
uniformly in all these choices. The new scale and normalization
satisfy
N^{1/2} <= M_0^{-k} N^{1 - k tau} <= n_B < N,
1 <= Lambda_B <= 1 + M_0^k N^{k tau - 1}. (6.7)
Proof. The scale bounds follow from D <= L_D <= D^k and M_0 >= 2;
their first inequality holds for large N because k tau < 1/2. In
particular, with Q' = n_B^beta,
H <= Q' <= Q. (6.8)
By the signed Holder inequality (Proposition 3.1) it is enough to prove
(6.6) for Z_{l, S}(g), where g is any one of the specialized inputs.
Step 1 (large-denominator truncation). Keep only the frequencies
of g with denominator at most H, and call the result g_{<= H}.
Telescope the diagonal integral one slot at a time. Lemma 3.3
(large-denominator estimate) combined with Proposition 3.4 (lift
moments) and Lemma 3.5 (specialization) bounds the change by
|Z_{l, S}(g) - Z_{l, S}(g_{<= H})|
<< H^{-gamma} q_B N^{o(1)} = O(N^{-sigma}). (6.9)
Indeed specialization costs q_B^{1/2} in the L^2 norm of the
removed part and q_B^{1/(2(d-1))} in each of the other d - 1 norms;
their product is q_B. All frequency selections retain whole exact
supports before specialization. The saving follows from
beta gamma / 2 - tau > sigma.
Step 2 (progressions and avoidance). The auxiliary-family identity
transports avoidance along progressions of step L_D = lambda(D).
Since D | L_D, split the prescribed class mod D into its L_D / D
classes mod L_D. Let a' mod D be the class determined by the
specialization. For j in [L_D] with j = a' mod D, put
c_j = j - L_D,
A'_j = { x' in [n_B] : L_D x' + c_j in A },
H_j = L_{[n_B], S, H} 1_{A'_j}.
Every integer of A in the class a' mod D has a unique
representation L_D x' + c_j. The definition excludes any points
beyond N.
Each A'_j avoids at lD. Indeed a nonzero difference h_{lD}(m') in
A' would give the difference lambda(D) h_{lD}(m') = h_l(D m' - i) in
A, with 0 <= i < D supplied by Lemma 1.1(c). We have D m' - i >= 1
and T_l(D m' - i) = T_{lD}(m'); the difference is nonzero because
lambda(D) > 0. This violates avoidance of A at l.
Step 3 (exact fiber decomposition). Since q_B H <= Q, Lemma 6.1
gives the exact identity
g_{<= H}(z) = (Lambda_B / (L_D / D))
sum_{j in [L_D], j = a' mod D} H_j(L_D^{-1}(z - c_j)). (6.10)
Every prime divisor of L_D divides D, so multiplication by L_D is
invertible on X_S and preserves Fourier denominators. For q(xi) <= H
the coefficient of the jth summand at xi is
hat H_j(L_D xi) e(-c_j xi); summing over j and multiplying by
Lambda_B / (L_D / D) = D n_B / N gives exactly (6.5) for I = [N].
Both sides of (6.10) have zero coefficients at denominators > H.
Step 4 (seminorm contraction). There are L_D/D summands. Lemma 3.2
(seminorm triangle inequality) and its transport identity (3.4)
give
Z_{l, S}(g_{<= H}) <= Lambda_B^d max_j Z_{l D, S}(H_j). (6.11)
Raise the cutoff in H_j from H to Q'. The same large-denominator
argument, now without specialization, changes the diagonal integral
by O(H^{-gamma} N^{o(1)}) = O(N^{-sigma}). The moment hypothesis
holds at the new scale because n_B >= 10(Q')^6 for large N
(by 6 beta < 1 and (6.7)).
Step 5 (extend and identify). The extended lift is obtained from
L_{[n_B], P_{n_B}, Q'} 1_{A'_j} by averaging out its coordinates in
B cap P_{n_B} and adding unused coordinates in S \ P_{n_B}. Averaging
contracts the seminorm and adding unused coordinates does not change
it (Lemma 3.2). So its diagonal integral is at most
Y_{l D}(n_B, A'_j) <= Y_{l D}(n_B). Combining with (6.9) and (6.11)
proves the diagonal bound, hence (6.6). None of these steps
restricts the size of l. QED
6.5. Cancellation for good residue assignments.
The preceding proposition controls every assignment in (6.4). We
now get a stronger bound for those assignments whose cycle
increments can occur as local polynomial values.
Definition. Call (s_v)_{v in V} good at l if every cycle increment
s_{v_{j+1}} - s_{v_j} (0 <= j < L) is admissible mod M_0 at l. Let
rho_l be the proportion of good assignments. The zero-sum admissible
sequences of Lemma 1.5 give
rho_l >= rho_* := M_0^{1 - L} > 0. (6.12)
Indeed combine such a sequence at each prime power dividing M_0 by
the Chinese remainder theorem: after prescribing these L increments,
the initial cycle residue and every residue off the cycle are free.
Proposition 6.3 (Cancellation on nonexceptional laws). For all
sufficiently large N, all 1 <= l <= N^zeta, every A subset of [N]
avoiding at l, and every good assignment (s_v)_{v in V},
| int prod_{v in V} G_{s_v}(z_v)
d(tensor_p (mu_{l,p} - eps_{l,p})) |
<< N^{-sigma}. (6.13)
The implied constant is uniform in l, A, and the assignment.
Proof. The aim is to leave just the two endpoint lifts on an ordered
cycle edge. First average over short intervals; then truncate and
expand the remaining inputs. Each expansion term uses prime
coordinates with small product, and conditioning on those coordinates
leaves the pair to which Proposition 5.1 applies.
Step 1 (short intervals). It suffices to use the probability
measures mu_{l,p}^circ = (mu_{l,p} - eps_{l,p})/(1 - eta_{l,p}),
since restoring the omitted masses multiplies the integral by
prod_p (1 - eta_{l,p}) <= 1. Partition [N] into m = floor(N^tau)
consecutive intervals of lengths as equal as possible. Each G_s is
the weighted average of L_{I, P_N, Q} f_s over these intervals,
with weights |I|/N. So the integral is an average over independently
chosen intervals for all slots. Their lengths are comparable to
N^{1 - tau}, so the lift moment bound applies (6 beta + tau < 1).
Ordinary Holder and uniform marginals bound every interval
assignment by N^{o(1)}.
The probability that all L cycle slots use the same interval is
sum_I (|I|/N)^L = O(N^{-(L-1) tau}). These assignments contribute
O(N^{-sigma}). For any other assignment, the cyclic sequence of
interval indices is nonconstant and has a strict increase along
some directed cycle edge. Choose such an edge, with first endpoint
in I_1 and second endpoint in a later interval I_2.
Step 2 (truncation and expansion). Keep the two endpoint lifts
unchanged, and truncate every other lift to denominators at most
T = N^tau. Telescoping with Lemma 3.3 and the lift moment bound
costs O(T^{-gamma} N^{o(1)}) = O(N^{-sigma}). Expand the truncated
d - 2 inputs into characters: coefficients of absolute value at
most M_0, each input has O(T^2) frequencies, so the sum of absolute
coefficients in the product expansion is O(T^{2(d-2)}).
Step 3 (fix one expansion term). Let B be the union of the prime
supports of the chosen characters, and put
R = q_B <= T^{d-2}, D = M_0 R, S = P_N \ B.
Condition on the entire tuple of labels at every prime in B, with
any conditioning of positive probability under the product of the
mu_{l,p}^circ. The expanded characters become constants of modulus
one. The specialized endpoint lifts give functions g_1, g_2 on
X_S, and the laws at primes in S remain unchanged. By translation
invariance, the pair integral is
sum_{xi on X_S} hat g_1(-xi) hat g_2(xi) Gamma(xi), (6.14)
where Gamma is the multiplier of this directed cycle edge. Indeed
only opposite characters survive common translation, and their
product is the character of the second label minus the first.
By (5.1) (edge multiplier bound), Parseval, and the specialized
moment bound, the part with q(xi) > H is at most
H^{-gamma} ||g_1||_2 ||g_2||_2 << H^{-gamma} R N^{o(1)}. (6.15)
For the remaining frequencies H R <= Q and D <= N^beta for large N
(since (d - 2) tau < beta/2). Apply Lemma 6.1 (residue
specialization) to the endpoint lifts, obtaining classes
a'_1, a'_2 mod D. Their directed difference is admissible at l:
goodness supplies admissibility mod M_0, and the cycle property of
mu_{l,p}^circ supplies it at the conditioned primes p in B. Set
J_i = 1_A 1_{I_i} 1_{x = a'_i mod D}, i = 1, 2,
as functions on [N]. Every contributing pair has x < y because I_1
precedes I_2, and avoidance gives y - x not in D_l. All hypotheses
of Proposition 5.1 hold.
Its frequency set Xi is exactly the remaining frequencies on X_S
with denominator at most H: coprimality with D excludes the primes
below P and the primes in B; every prime in a denominator at most H
is at most Q. The coefficient identity becomes
hat g_i(xi) = (D N / |I_i|) hat (J_i)_{[N]}(xi) for q(xi) <= H.
Consequently (5.2) bounds the low-denominator part of (6.14) by
O(H^{-gamma/2} D N^2 / (|I_1| |I_2|))
= O(H^{-gamma/2} D N^{2 tau}). (6.16)
After multiplication by the expansion bound O(T^{2(d-2)}), both
(6.15) and (6.16) are at most
N^{-beta gamma / 4 + (3d+2) tau + o(1)} = O(N^{-sigma}),
because beta gamma / 4 > (3d + 2) tau + sigma by (5.1). The
estimates are uniform in the conditioning, the selected edge, and
the interval assignment. Averaging, and including the earlier
truncation and coincident-interval errors, proves (6.13). QED
6.6. The contracting recurrence.
We have cancellation after removing the exceptional part of every
local law. To return to the full laws in (6.4), expand that
removal. The total mass of the resulting exceptional terms is small
enough to pay for their possibly shorter progression scales.
For B subset of P_N write
eta_{l, B} = prod_{p in B} eta_{l,p},
D_B = M_0 q_B, n_B = ceil(N / lambda(D_B)).
Thus D_empty = M_0. The mass estimate of Proposition 2.1 and the
choice of P (Section 2.8) give, uniformly in l,
sum_{B nonempty subset of P_N} q_B eta_{l, B}
<= prod_{p >= P} (1 + p^{-3}) - 1 <= 1/4. (6.17)
Proposition 6.4 (Recurrence). There is a constant C_1 such that
for every sufficiently large N and 1 <= l <= N^zeta,
Y_l(N) <= C_1 N^{-sigma} + (1 + u_N) [(1 - rho_l)
Y_{l M_0}(n_empty)
+ rho_l sum_{B nonempty, q_B <= N^tau} eta_{l, B}
Y_{l D_B}(n_B)], (6.18)
u_N = (1 + M_0^k N^{k tau - 1})^d - 1 = o(1). (6.19)
The constant and the sufficiently-large threshold are independent
of l.
Proof. Fix an avoiding set A. For assignments in (6.4) that are not
good, apply Proposition 6.2 with B = empty. Their proportion is
1 - rho_l. For a good assignment, Proposition 6.3 handles the
product of the measures mu_{l,p} - eps_{l,p}. The difference from
the full product is
tensor_p mu_{l,p} - tensor_p (mu_{l,p} - eps_{l,p})
= sum_{B nonempty} (-1)^{|B|+1}
(tensor_{p in B} eps_{l,p})
tensor (tensor_{p notin B} mu_{l,p}).
The Bth summand has positive mass eta_{l, B}. If this mass is zero
the summand vanishes. Otherwise, after normalization, every
one-slot marginal is uniform by simultaneous translation invariance.
Holder and Proposition 3.4 bound the integral by N^{o(1)} in
absolute value. The terms with q_B > N^tau have total mass at most
N^{-tau}/4 by (6.17), and hence contribute O(N^{-sigma}).
For q_B <= N^tau, condition on all labels at the primes in B. The
remaining law is the product of the full measures mu_{l,p}, so
Proposition 6.2 applies to the specialized inputs. Its factor
Lambda_B^d is at most 1 + u_N by (6.7), and its errors sum to
O(N^{-sigma}) by (6.17). Taking absolute upper bounds in the signed
expansion and averaging over the good assignments gives (6.18).
Finally maximize over A in (6.4). QED
6.7. Uniform power decay.
The recurrence is useful only when l <= N^zeta, but its successors
l D_B need not be in the analytic range at n_B. We therefore prove
the bound simultaneously for every auxiliary parameter l. The factor
l^{b_0} below covers the complementary range via the elementary
bound (6.3).
Proposition 6.5 (Uniform power decay). There are constants
a_0, b_0 > 0 and C_0 >= 1, depending only on h, such that for all
integers l, N >= 1,
Y_l(N) <= C_0 l^{b_0} N^{-a_0}. (6.20)
Proof. After all parameters in (5.1) are fixed, choose a_0 > 0
sufficiently small and put b_0 = 2 a_0 / zeta, so that
a_0 < sigma, s_0 := k a_0 + b_0 <= 1,
theta_0 := M_0^{s_0} (1 - 3 rho_* / 4) < 1. (6.21)
This is possible because s_0 -> 0 as a_0 -> 0 while rho_* > 0 is
fixed.
If l > N^zeta, then l^{b_0} N^{-a_0} >= N^{a_0}. The trivial
estimate (6.3), used with exponent a_0, proves (6.20) in this range
after a fixed enlargement of C_0. This handles all l > N^zeta
before the induction.
Remaining range. Induct on the integer N, simultaneously for all
l >= 1. At a sufficiently large N with l <= N^zeta, every
successor in (6.18) satisfies n_B < N by (6.7). The induction
hypothesis is available at l D_B, whether or not l D_B <= n_B^zeta.
Since n_B >= N / D_B^k, it gives
Y_{l D_B}(n_B)
<= C_0 (l D_B)^{b_0} n_B^{-a_0}
<= C_0 l^{b_0} N^{-a_0} D_B^{s_0}.
Use q_B^{s_0} <= q_B and (6.17) to bound the bracket in (6.18):
C_0 l^{b_0} N^{-a_0} M_0^{s_0} (1 - rho_l + rho_l/4)
<= C_0 l^{b_0} N^{-a_0} theta_0,
where (6.12) was used in the last step. Choose a fixed threshold
N_*, large enough for the recurrence and so that
C_1 N^{a_0 - sigma} + (1 + u_N) theta_0 <= 1 (N >= N_*).
Such N_* exists by (6.21) and u_N = o(1); it is independent of
C_0 >= 1. Dividing the recurrence by C_0 l^{b_0} N^{-a_0} proves the
induction step, since C_0 l^{b_0} >= 1.
Finally enlarge C_0 to cover both the previously treated range
l > N^zeta and every N < N_*, uniformly in l, by (6.3). This
completes the induction. QED
6.8. Proof of Theorem 0.2.
At l = 1, T_1(m) = m and h_1 = h. So the hypothesis of the theorem
is exactly avoidance at l = 1. Combining (6.2) with (6.20),
(|A| / N)^d <= Y_1(N) <= C_0 N^{-a_0},
|A| <= C_0^{1/d} N^{1 - a_0/d}.
Thus we may take
c_h = a_0 / d, C_h = C_0^{1/d},
for every N >= 1. Only strictly ordered pairs (and hence positive
differences) were used in cancellation. Restriction to progressions
preserved nonzero forbidden values and their prime arguments
(Lemma 1.2). A value h(p) = 0 therefore makes no additional demand
at any stage. QED
--------------------------------------------------------------------
Closing remarks
--------------------------------------------------------------------
On constants. All constants c_h and C_h in Theorem 0.2 depend only
on h (via its degree k, its coefficients through the p-adic roots
z_p, and the thresholds P_1, x_*, N_*). The exponents satisfy
c_h = a_0 / d, 1/d = K^{-w}, K = L!, w = 2 t + 1,
a_0 in (0, sigma),
with sigma = gamma tau / 4, tau = beta gamma / (1000 k (d+1)),
beta = nu/100, nu = min(eta, c_m/k)/10, eta = 1/(10000 k), and
c_m = c_m(k, eta/4) from Proposition 4.2. The saving is extremely
small but positive; only its existence matters.
On the prime-argument condition. The unit-root condition at every
prime power (prime-intersectivity) enters at three points:
- the p-adic root z_p and the auxiliary polynomial h_l
(Section 1.1);
- the "admissible" local values from Lemma 1.5 and the
zero-sum cycle of length L (Sections 1.3-1.5);
- the signed kernel k_1 supported on prime arguments of h_l and
the local realization (5.14) via unit means (Section 5.6).
If we drop the unit-root condition at some modulus, these three
places break down (no admissible cycle exists at that modulus), and
the argument gives no control.
On future improvements. The saving is fixed but tiny; it is
plausible that more careful choices of t, w, L, and the parameter
hierarchy yield larger exponents. The skeleton of the argument
(reflection-positive tuple laws + Fourier lifts + prime-supported
kernel) seems flexible; optimizing the constants would be a project
in itself. Only the structure - existence of some c_h > 0 - is
established here.