A lattice in R^n is a discrete subgroup spanned by n linearly
independent vectors. It is unimodular if its fundamental
parallelepiped has volume one. Modulo Z-basis change the space
of unimodular lattices is the homogeneous space X_n = SL_n(Z) \
SL_n(R); it carries a unique left-SL_n(R)-invariant probability
m_n, the Haar measure, and a right action of the diagonal group
A_n of determinant-one diagonal matrices.
Fix a prime n >= 5, a totally real number field K of degree n,
a full Z-lattice M in K, and an ordering sigma of the real
embeddings of K. This data determines a finite packet of closed
diagonal A_n-orbits in X_n, one per K^*-homothety class of M
and per sign pattern in the diagonal +-1 matrix group W_n whose
product of diagonal entries is one. The volume-weighted packet
probability, written mu_{K,M,sigma}, is the arithmetic object
of study.
The multiplier-order discriminant
D(M) = |Disc(K)| [O_K : O(M)]^2 measures the arithmetic size
of M. The central theorem: for every fixed prime n >= 5 and
every sequence of data with D(M) tending to infinity, the
packet probability mu_{K,M,sigma} converges weakly to m_n. This
is the higher-dimensional Duke equidistribution theorem for
prime degree, established for every prescribed local homothety
type of M, including lattices over nonmaximal orders of
arbitrary index. Tightness is part of the conclusion: mass
concentrates on compacts, nothing leaks into the cusp of X_n.
https://doi.org/10.1007/BF01393993
The proof has four layers. An exact unfolding identity expresses
the packet average of any lattice-vector count as a weighted sum
over integral ideals of K. A residue-relative weighted count of
ideals in short intervals, uniform in the local data, supplies
the arithmetic input. A tightness-plus-ball-bound package forces
positive entropy in the flow direction. The prime-degree measure
classification of Einsiedler, Katok, and Lindenstrauss then
promotes positive entropy to Haar. The arithmetic engine is an
effective residue estimate for the Dedekind zeta function, in
the exceptional-zero tradition of Stark, together with Shiu's
short-interval theorem for multiplicative functions.
./ekl2006.pdf
https://doi.org/10.1007/BF01405166
https://doi.org/10.1515/crll.1980.313.161
A lattice in R^n is the set of integer linear combinations of
n linearly independent vectors v_1, ..., v_n:
L = Z v_1 + Z v_2 + ... + Z v_n.
The lattice L is unimodular if |det(v_1 | v_2 | ... | v_n)| = 1;
one says its fundamental parallelepiped has covolume one.
Two unimodular lattices that differ only by a Z-basis change
are considered the same point of X_n. Formally,
X_n = SL_n(Z) \ SL_n(R),
with the right action of SL_n(R) given by
(Z^n g) h = Z^n (g h).
This is a noncompact, finite-volume manifold of dimension
n^2 - 1. It carries a unique left-SL_n(R)-invariant probability
measure, called the Haar probability and denoted m_n. It is the
uniform probability: for every continuous compactly supported
function f on X_n, the packet probabilities mu_i converge to
m_n exactly when int f d mu_i -> int f d m_n.
The space X_n is the moduli space of covolume-one lattices in
R^n up to Z-basis change: for g in SL_n(R), the left coset
SL_n(Z) g is the unimodular lattice with rows given by the
rows of g.
The diagonal group
A_n = { diag(e^{u_1}, ..., e^{u_n}) :
u_1, ..., u_n in R, u_1 + ... + u_n = 0 }.
This is a connected abelian Lie group isomorphic to R^{n-1}.
It acts on X_n by right multiplication: (Z^n g) a = Z^n (g a).
For Lambda in X_n and a in A_n, write Lambda a for the image
of Lambda under this action.
The orbits Lambda A_n are compact when Lambda is an arithmetic
lattice attached to a number field (the only case of interest). The classical Duke theorem, in this language, says
that for n = 2 certain infinite families of these closed A_n-
orbits become distributed like m_n. The higher-rank Duke
problem asks for the same phenomenon for n >= 3.
The following data is fixed throughout:
- n >= 5 is a fixed prime.
- K is a totally real number field of degree n. "Totally
real" means every embedding K -> C has image in R; it
has exactly n such embeddings, and an ordering
sigma = (sigma_1, ..., sigma_n) is fixed.
- M is a full Z-lattice in K: a free Z-submodule of K of
rank n, whose Q-span is all of K.
- The multiplier order of M is
O(M) = { alpha in K : alpha M is contained in M }.
This is an order of K; it always contains the maximal
order R = O_K.
- The multiplier-order discriminant is
D(M) = |Disc(K)| * [O_K : O(M)]^2,
where [O_K : O(M)] is the index. When M is an R-ideal this
equals |Disc(K)|.
For each rational prime l, localize at l: K_l = K (x)_Q Q_l,
M_l = M (x)_Z Z_l, etc. Two full lattices M' and M'' in K
are locally homothetic if, for every l,
M'_l = c_l M''_l in K_l, with c_l in K_l^*.
That is, the two completions agree at every prime after a
scalar multiplication. This is strictly finer than "same
multiplier order": two lattices can have the same O(M) and D(M)
while lying in different local types.
Fix the local homothety type of M (the family of all full
lattices locally homothetic to M). For each K^*-homothety
class M' in this local type, embed M' in R^n via sigma and
rescale to covolume one:
Lambda_{M', sigma}
= covol(sigma(M'))^{-1/n} sigma(M').
Let W_n be the finite group
W_n = { diag(eps_1, ..., eps_n) : eps_j in {+1, -1},
eps_1 * eps_2 * ... * eps_n = 1 }.
W_n has order 2^{n-1}; it is the group of sign matrices that
preserve lattice shapes up to sign reversal. Since n is odd,
the sign "flip all coordinates" is in W_n; this is the source
of the "distinct orbits" bookkeeping below.
The packet of (K, M, sigma) is the finite set of orbits
P_{K, M, sigma} = { Lambda_{M', sigma} w A_n :
M' runs over K^*-homothety classes in the local type,
w runs over W_n,
with duplicate orbits identified }.
(The next part proves both compactness and finiteness; the
"packet model" lemma reconstructs the packet as a disjoint
union, orbit by orbit, from the ideal-class and local-unit
data.)
For an orbit O in the packet, let vol(O) be the volume with
respect to Haar measure on A_n, normalized by product Haar on
A_n, and let nu_O be the A_n-invariant probability measure on O.
Then the packet probability is
mu_{K, M, sigma} =
( sum_{O in P_{K, M, sigma}} vol(O) nu_O )
/ ( sum_{O in P} vol(O) ).
By construction mu_{K, M, sigma} is a probability on X_n. It
is a weighted average of the Haar probabilities of the packet
orbits, weighted by orbit length.
THEOREM (main).
Fix a prime n >= 5, and a sequence (K_i, M_i, sigma_i)
satisfying D(M_i) -> infinity. Then in the weak-* topology of
probability measures on X_n,
mu_{K_i, M_i, sigma_i} --> m_n.
Equivalently: for every continuous compactly supported f on X_n,
int f d mu_{K_i, M_i, sigma_i} -> int f d m_n.
Additionally the family is tight: no amount of mass escapes
into the cusp of X_n. Weak convergence follows from tightness
plus identification of every subsequential limit.
The theorem is unconditional, exact for every prescribed local
type, for every totally real field in the allowed degrees and
every full lattice M in that field. In particular, D(M) may
grow while |Disc(K)| stays bounded, if the index [O_K : O(M)]
grows.
- LINNIK (1968) initiated the ergodic approach to arithmetic
equidistribution.
https://link.springer.com/book/9783642866333
- DUKE (1988) proved equidistribution of length-weighted
closed geodesics attached to fundamental real-quadratic
discriminants on the modular surface.
- EINSIEDLER, LINDENSTRAUSS, MICHEL, VENKATESH ("ELMV";
2009, 2011) built the general relation "arithmetic
separation plus subconvexity" controlling periodic torus
orbits, and proved cubic equidistribution unconditionally.
The ELMV packet construction already distinguishes local
homothety from the coarser condition "same multiplier
order."
./elmv2009.pdf
./elmv2011.pdf
- KHAYUTIN (2019) obtained the entropy side using arithmetic
separation and Galois invariants, under maximal-order and
two-transitivity hypotheses.
./khayutin2019.pdf
- LEMKE OLIVER, THORNER, ZAMAN (2024) proved
equidistribution for sequences of orders in every fixed
prime degree at least five, at the cost of a small
exceptional set of fields.
./lotz2024.pdf
- The theorem above: unconditional, no exceptional set, for
every local type, including sequences of orders with
increasing index in a fixed field.
The arithmetic ingredient is not subconvexity. It is an
effective residue estimate for the Dedekind zeta function
zeta_K together with the uniform form of Shiu's theorem for
multiplicative functions. Odd prime degree rules out the
quadratic subfields that appear in the exceptional alternative
of Stark's continuation of the zero-free region.
Put R = O_K, D = |Disc(K)|, q = [R M : M], Q = q * sqrt(D).
This is the natural counting scale; D(M) <= Q^{2n}. The rest
of the proof continues:
Step A. The packet measure of any "lattice-vector
count" E_f (sum of f over nonzero lattice vectors) equals a
specific weighted sum over integral ideals:
int E_f d mu = (1 / (sqrt(D) kappa_K))
* sum_a P(a) V_f (N(a) / Q).
P(a) is a containment probability for the prescribed
local type, V_f is a "product-fiber" integral (f integrated
along a fixed-product slice of the real diagonal torus), and
kappa_K = Res_{s=1} zeta_K(s).
Step B. The weights P(a) have exact total
mass Z_S / q and power-saving tails. The Dedekind zeta
residue estimate (via Stark) plus Shiu's theorem give a
weighted count of ideals in short intervals:
sum_{x - y < N(a) <= x} P(a) <<_n kappa_K y / q,
for Q^{3/4} <= x <= A_0 Q and eta x <= y <= x.
Step C. The weighted count gives two
geometric consequences:
(i) (Cusp.) The packet mass of the set of lattices with a
nonzero vector of sup-norm <= theta is at most
C_n theta^n. So the family is tight.
(ii) (Balls.) The weak limit mu on X_n satisfies an
ordinary-ball bound mu(x B(r)) <= C_Omega r^n for
x in any fixed compact Omega. This is one power of
r smaller than dim A_n = n-1 could give.
Step D. The ball bound forces positive entropy
in the flow direction a(1) in A_n. Ergodicity plus the
prime-degree measure classification theorem of EKL then
identifies every ergodic component of the limit as the
Haar measure m_n. Tightness and uniqueness of the limit
finish the proof.
Fix a Z-basis alpha_1, ..., alpha_n of M. The matrix
B_sigma = ( sigma_j(alpha_i) )_{i,j}
has (i, j) entry sigma_j(alpha_i). For alpha = sum_i alpha_i c_i
in M with integer coefficients c_i, the row vector of
embeddings is
sigma(alpha) = (sigma_1(alpha), ..., sigma_n(alpha))
= (c_1, ..., c_n) B_sigma.
Thus sigma(M) = Z^n B_sigma, a row lattice of rank n in R^n.
Its covolume is |det B_sigma| = sqrt(Disc(M)) = sqrt(D) [R:O(M)],
where Disc(M) is the discriminant of M. For a nonzero fractional
R-ideal I, covol(sigma(I)) = sqrt(D) N(I), with N(I) the
absolute norm of I.
Define
C = { t in (R^*)^n : prod_{j=1}^n |t_j| = 1 }.
Every t in C can be written uniquely as
t = (eps_1, ..., eps_n) * (e^{u_1}, ..., e^{u_{n-1}},
e^{-u_1 - ... - u_{n-1}}),
where eps_j in {+1, -1} and u_1, ..., u_{n-1} in R. The measure placed
on C is counting measure on the 2^n sign
choices, times Lebesgue measure on the u-coordinates. Under
the embedding c -> diag(c) of C into SL_n(R), this is the
Haar measure on the diagonal maximal split torus.
Pick real numbers lambda_1, ..., lambda_n with
lambda_1 + ... + lambda_n = 0 and lambda_i distinct. Define
a(t) = diag(e^{lambda_1 t}, ..., e^{lambda_n t}), t in R.
Let
lambda = min_{i != j} |lambda_i - lambda_j| > 0.
This "gap" lambda measures how quickly the diagonal flow
separates coordinates. Entropy of the flow a(1) will be
bounded below by a fixed multiple of this gap.
The packet sum is expressed next as a weighted ideal sum.
Put R = O_K, E = R^* = O_K^*, D = |Disc(K)|, and
kappa_K = Res_{s=1} zeta_K(s). Define the maximal-order span
of M and the index
J_0 = R M (an integral R-ideal containing M),
q = [J_0 : M] (the index),
Q = q * sqrt(D).
Claim II.1.1. q R is contained in O(M), and O(M) is contained
in R. In particular, D(M) = D [R : O(M)]^2 <= D q^{2n} <= Q^{2n}.
Proof. Because J_0 = R M and M has rank n, one has
q J_0 subset M (the quotient J_0/M has order q and is
annihilated by q; use the standard finite-abelian-group
structure). Multiply by R: q R J_0 = q J_0 subset M. But
q R J_0 = (q R) J_0, and every element of q R is an
algebraic integer in K, so q R is contained in the multiplier
order O(J_0) of J_0. Since J_0/M is finite and q annihilates
it, O(J_0) is also the multiplier order of M modulo q, and
combining with the definition yields q R subset O(M).
The reverse inclusion: any alpha in O(M) has an integral
action on M, so alpha is an algebraic integer in K, i.e., in
R. Thus O(M) subset R. Composing gives q R subset O(M) subset R,
hence [R : O(M)] <= [R : qR] = q^n, so D(M) <= D q^{2n}.
Finally, D(M) -> infinity implies Q -> infinity: from
D(M) <= Q^{2n}, D(M) -> infinity gives Q -> infinity. QED
For a rational prime l, set R_l = R (x)_Z Z_l, M_l = M (x)_Z
Z_l. The integral closure of the ideal J_0 at l is
(J_0)_l = j_l R_l for some j_l in K_l^*. Define
L_l = j_l^{-1} M_l, an additive subgroup of K_l.
Claim II.2.1. L_l is an open subgroup of R_l of finite index.
Moreover, R_l L_l = R_l, [R_l : L_l] = q_l, and q = prod_l q_l.
Proof. First, [R_l : j_l^{-1} M_l] = [j_l R_l : M_l] = (q)_l,
since J_0 / M has order q. That is finite, so L_l is open of
finite index. For "R_l L_l = R_l": R_l L_l = R_l j_l^{-1} M_l
= j_l^{-1} R_l M_l = j_l^{-1} (J_0)_l = j_l^{-1} j_l R_l = R_l.
The index computation [R_l : L_l] = q_l is immediate from the
finite-index formula. The product formula q = prod_l q_l is
the multiplicativity of the index over the S-supported primes.
QED
Set S = {l : l | q}. For l not in S, q_l = 1, so
L_l = R_l.
For each ordinary ideal class [I] of R, choose one fractional
R-ideal I in that class. Let N(I) denote the finite set of
full lattices N with
(a) N is locally homothetic to M;
(b) R N = I.
Claim II.3.1. The set N(I) is finite. Every N in N(I)
satisfies [I : N] = q and lies between q I and I. The
cardinality T = |N(I)| is independent of [I]. Local unit Haar
probability on each finite set of local choices induces uniform
probability on N(I), and the choices at different primes are
independent.
Proof. Write I_l = i_l R_l for some i_l in K_l^*. By II.2, the
possible completions of a member of N(I) are exactly
N_l = i_l z_l L_l, with z_l in R_l^*. Taking maximal-order
spans in the local homothety forces the remaining multiplier to
be a unit: if N'_l = c_l M'_l, then (R N')_l = c_l (R M')_l
= c_l J_0_l = (c_l R_l) J_0_l, so the extra multiplier c_l
must be a unit modulo the local maximal-order normalizer.
Reversing, any (z_l) in prod_l R_l^* produces a candidate
lattice N_l = i_l z_l L_l, and these are contained in I_l (since
L_l subset R_l) and contain q I_l (since q_l R_l subset L_l;
similarly L_l subset R_l applied to q). Finiteness: N(I) is a
subset of the finite set of lattices between q I and I.
The patching over S is unique by the primary decomposition of
I/qI. The cardinality T = |N(I)| is independent of [I]:
multiplication by i_l identifies the finite set of local unit
orbits at l for two choices of I, so the count is independent
of the ideal. Local unit Haar probability gives each choice
probability 1/|R_l^* : stab_{R_l^*}(L_l)|; the product over l
in S of these probabilities is uniform on N(I), independent of
the choice of I. QED
One caution: the label count T need not equal q. T counts
unit translates, q is the base saturation index; the two
coincide only when q_l = |R_l^* : stab_{R_l^*}(L_l)| at every
l in S, which is not the general case.
Recall C = {t in (R^*)^n : prod |t_j| = 1} with half-log-Haar
measure from I.2. The embedded unit group sigma(E) is a
discrete cocompact subgroup of C. Let F be a measurable
fundamental domain. By the totally real case of the analytic
class-number formula,
vol(F) = 2^{n-1} R_K, and h_K vol(F) = sqrt(D) kappa_K.
h_K is the ordinary class number, R_K the regulator
expressed in deleted logarithmic coordinates. The 2^{n-1}
accounts for the 2^n sign choices in C modulo the 2 sign
kernel of log|.|.
LEMMA II.5.1 (packet model).
The map
Phi(N, t) = (Q N(I))^{-1/n} sigma(N) t,
defined on pairs (N, t) in N(I) x C for each ideal class [I],
induces a bijection
disjoint union over [I] of ( N(I) x C ) / E
onto the union of the packet orbits. The group E acts by
eps : (N, t) -> (eps N, sigma(eps)^{-1} t).
Counting measure on N(I)-labels times half-log-Haar on C
descends to the sum of the orbit Haar measures. In particular,
the packet probability is uniform over the ideal classes [I]
and over N in N(I), followed by integration dt / vol(F) over F.
Proof. SURJECTIVITY. Every packet orbit is, by definition,
Lambda_{M', sigma} times some t in C. The maximal-order span
of M' is a fractional ideal in some class [I]; local
homothety plus the choice of representative N in N(I) gives
the label data, and the real scaling gives t. So Phi hits
every orbit.
INJECTIVITY. Suppose two pairs (N, t) and (N', t') map to
the same point of X_n:
(Q N(I))^{-1/n} sigma(N) t
= (Q N(I'))^{-1/n} sigma(N') t'.
Absorb the ratio of the two normalizing scalars into a single
diagonal real map D: then D sigma(N) = sigma(N').
CLAIM: N' = bN for some b in K^*. Proof of claim: first pick
alpha in N with all n coordinates of sigma(alpha) nonzero.
Such alpha exists because N is a full lattice: for any Z-basis
x_1, ..., x_n of N, the integer linear combination
alpha = x_1 + N x_2 + N^2 x_3 + ... + N^{n-1} x_n
has coordinates that are polynomials in N, and each coordinate
is not the zero polynomial because the x_i are K-linearly
independent; so only finitely many integers N give a zero
coordinate, and all other N give coordinates all nonzero.)
Now: D sigma(alpha) = sigma(beta) for some beta in N' (because
D sigma(N) = sigma(N')). Since sigma(alpha) has no zero
coordinate and D is diagonal, D = diag(sigma(beta / alpha)).
In particular D = sigma(b) for b = beta / alpha in K^*. So
sigma(N') = sigma(b) sigma(N) = sigma(b N), hence N' = b N.
(Without the "all coordinates nonzero" choice of alpha, the
diagonal D would not be uniquely determined; this is exactly
the place where the choice matters.)
By construction R N = R N' = I. So b I = I, hence
b in E = R^*. The pairs (N, t) and (N', t') differ by the
E-action, establishing injectivity of the quotient map.
STABILIZER AND VOLUME. The stabilizer of a fixed label (N, +1)
in E is the totally positive subgroup of
{eps in E : eps N = N}, of finite index in E. Its image in C
acts by log translations. The all-signs-negative unit -1 acts
trivially on C. So the E-action is faithful on log coordinates,
and the quotient of each label orbit by its stabilizer is a
single compact A_n-orbit. By injectivity, distinct label
orbits produce distinct packet orbits (no double counting).
Each orbit inherits precisely its Haar probability, and the
total measure of the "fundamental domains" N(I) x F is
h_K * T * vol(F). Dividing by this quantity gives the packet
probability: uniform on the classes [I], uniform on N(I)
within each class, integrate dt / vol(F) over F. QED
For a rational prime l and a tuple k_l = (k_p)_{p | l} of
nonnegative integers, let
P_l(k_l) = Pr_{z in R_l^*} { x in z L_l }.
Let x in K_l be a fixed element with valuations v_p(x) = k_p
for every p | l, and the probability is with respect to
normalized unit Haar measure on R_l^*. The probability does not
depend on the choice of x because the unit group acts
transitively on every fixed valuation shell.
For an integral nonzero R-ideal a, set
P(a) = prod_{l in S} P_l( (v_p(a))_{p | l} ).
Claim II.6.1. For alpha in I \ {0}, the probability that a
uniform N in N(I) contains alpha equals P( (alpha) I^{-1} ).
Proof. By Claim II.3.1, the N in N(I) correspond to
choices of z_l in R_l^* modulo unit stabilizer, with
N_l = i_l z_l L_l. The vector alpha is in N if and only if
alpha is in i_l z_l L_l for every l, equivalently (alpha / i_l)
is in z_l L_l; the probability at l is exactly P_l at the
valuation profile of alpha I^{-1}, and the product across l
gives P((alpha) I^{-1}). QED
For a nonnegative Borel function f on R^n and a unimodular
lattice Lambda in X_n, define
E_f(Lambda) = sum_{0 != v in Lambda} f(v).
For z > 0, define the product-fiber integral
V_f(z) = sum_{omega in {+-1}^n}
int_{R^{n-1}} f( omega_1 e^{u_1}, ...,
omega_{n-1} e^{u_{n-1}},
omega_n z e^{-u_1 - ... - u_{n-1}} )
du_1 ... du_{n-1}.
This is f integrated over the C-orbit with absolute coordinate
product z, in the half-log coordinates.
PROPOSITION II.7.1 (unfolding).
Let mu_* = mu_{K, M, sigma}. Then, for every nonnegative
extended-valued f,
int_{X_n} E_f d mu_* =
(1 / (sqrt(D) kappa_K)) *
sum_{a integral, a != 0} P(a) V_f( N(a) / Q ).
The identity holds in the sense of nonnegative extended
integrals; possible infinities agree.
Proof. By Lemma II.5.1,
int_{X_n} E_f d mu_*
= (1 / (h_K vol(F))) sum_{[I]} int_F
sum_{alpha in I \ {0}} P((alpha) I^{-1})
* f( (Q N(I))^{-1/n} sigma(alpha) t ) dt.
The map alpha -> (alpha) I^{-1} runs over the integral ideals
in the inverse class of I, each fiber being one free E-orbit.
Summing over E and using the fundamental domain F unfolds to
integration over C, by Tonelli. The absolute coordinate product
of the normalized vector (Q N(I))^{-1/n} sigma(alpha) is
N((alpha) I^{-1}) / Q = N( (alpha) I^{-1} ) / Q.
Translating the first n-1 logarithms and relabeling the 2^n
signs identifies this C-integral with V_f at height
N(a) / Q where a = (alpha) I^{-1}. Jacobian one (the
translation of log coordinates is measure-preserving and
relabeling is a bijection on sign patterns). Every integral
ideal a occurs in exactly one inverse class. The coefficient
is 1 / (sqrt(D) kappa_K), using h_K vol(F) = sqrt(D) kappa_K
from II.4. This proves the identity. QED
When M is an R-ideal, q = 1 and P(a) = 1, so the proposition
reduces to the classical Hecke unfolding formula for the
maximal order.
The local weights come first.
Let k = (k_l)_{l in S} range over tuples with k_l = (k_p)_{p | l},
each k_p a nonnegative integer. Set
d_l(k_l) = l^{ sum_{p | l} f_p k_p },
d(k) = prod_{l in S} d_l(k_l),
P(k) = prod_{l in S} P_l(k_l),
Z_l = prod_{p | l} (1 - l^{-f_p})^{-1},
Z_S = prod_{l in S} Z_l.
The symbols e_p and f_p are the ramification index and residue degree of
p over l in K / Q, so sum_{p | l} e_p f_p = n. Empty products are
one. Thus d(k) equals the norm of the S-supported part of an
integral ideal with profile k; P(k) is the containment
probability of such a profile.
LEMMA III.1 (shell-sum).
sum_k P(k) / d(k) = Z_S / q.
Proof. Normalize additive Haar measure on R_l to have mass one.
The valuation shell { x in R_l : v_p(x) = k_p for each p | l }
has additive mass Z_l^{-1} / d_l(k_l). (The local zeta factor
gives the normalization, and prod_p l^{-f_p k_p} = d_l(k_l)^{-1}
gives the shell volume.)
The conditional law on the shell is unit-Haar invariant because
the unit group is transitive on any fixed nonnegative valuation
shell. So P_l(k_l) = fraction of the k_l shell lying inside L_l.
Since L_l is an additive subgroup of R_l of index q_l, its
additive mass is 1 / q_l. Summing over shells:
sum_{k_l} P_l(k_l) / d_l(k_l) = Z_l / q_l.
Multiplying over l in S:
sum_k P(k)/d(k) = (prod Z_l) / (prod q_l) = Z_S / q.
Points with a zero component have additive measure zero in R_l
and contribute nothing. QED.
LEMMA III.2 (weight-moment).
Let s = 1/(2n). Then for every eps > 0,
sum_k P(k) d(k)^{s-1} <<_{n, eps} q^{eps} / q.
Proof. Choose x from additive-Haar on L_l; write its component
valuations as v_p = v_p(x). The key geometric fact is that
R_l L_l = R_l. This implies that the projection of L_l onto
each component K_p (the p-adic completion of K) contains a unit
of the local ring R_p. Modulo the kth power of the maximal ideal
of R_p, that unit produces at least l^{ceil(k/e_p)} distinct
residues. Since uniform additive measure on L_l descends to
uniform measure on this finite additive image, one gets
Pr_{x in L_l} ( v_p(x) >= k ) <= l^{-ceil(k/e_p)} for k >= 1.
Note: this argument does not need the projections to be
independent, or even to contain full integer rings; a unit per
component suffices.
Now put j = max_{p | l} ceil(v_p / e_p). Then
d_l(v_l) = l^{sum_p f_p v_p} <= l^{n j}. If j >= a >= 1, some
v_p is at least e_p (a - 1) + 1. By union bound and the
Pr inequality,
Pr(j >= a) <= n l^{-a}.
Therefore
E_{L_l} d_l(v_l)^s
<= 1 + n sum_{a >= 1} l^{(n s - 1) a}
<= 1 + C_n l^{-1/2},
because n s - 1 = 1/2 - 1 = -1/2 < 0.
Now the shell-mass computation again: on the kth shell, the
conditional law is uniform measure on L_l intersected with the
shell. So
sum_k P(k) d(k)^{s-1}
= (1/q) prod_{l in S} ( Z_l * E_{L_l} d_l(v_l)^s ).
Bound each factor:
Z_l * E d_l^s <= (1 - l^{-1})^{-n} (1 + C_n l^{-1/2}).
At every sufficiently large prime l (larger than a constant
depending only on n, eps), this factor is at most l^{eps}. The
finitely many remaining primes contribute a constant depending
only on n, eps. Since prod_{l in S} l <= q,
prod_{l in S} [ factor at l ] <<_{n, eps} prod_{l in S} l^{eps}
<= q^{eps}.
So sum_k P(k) d(k)^{s-1} <<_{n,eps} q^{eps}/q. QED.
LEMMA III.3 (weight-tail).
With s = 1/(2n),
sum_{d(k) > Q^{1/8}} P(k)/d(k) <<_n (1/q) Q^{-s/16}.
Proof. Apply Lemma III.2 with eps = s/16. For d(k) > Q^{1/8},
d(k)^{-1} = d(k)^{-s} * d(k)^{s-1} <= Q^{-s/8} d(k)^{s-1}.
So
sum_{d(k) > Q^{1/8}} P(k)/d(k)
<= Q^{-s/8} sum_k P(k) d(k)^{s-1}
<<_{n} (q^{-1} q^{s/16}) Q^{-s/8}
<= q^{-1} Q^{-s/8 + s/16} (using q <= Q)
= q^{-1} Q^{-s/16}.
QED.
The ideal count comes next.
Let a_K(m) = #{ integral R-ideals a : N(a) = m }. The residue at
s = 1 is
kappa_K = Res_{s=1} zeta_K(s)
= 2^{r_1}(2 pi)^{r_2} h_K R_K / (w_K sqrt(D)).
For K totally real, r_1 = n and r_2 = 0. Roots of unity: since
K is totally real, the only roots of unity are {+1, -1}, so
w_K = 2. Therefore
kappa_K = 2^{n-1} h_K R_K / sqrt(D).
Since vol(F) = 2^{n-1} R_K, this yields
h_K vol(F) = sqrt(D) kappa_K as claimed.
Throughout, uniforms: every constant subscripted by n is allowed
to depend only on n. No dependence on K, M, or
the local type is permitted. This is where the difficulty lies; estimates
must be uniform across all sequences of (K_i, M_i, sigma_i).
LEMMA IV.1 (residue).
For D sufficiently large in terms of n,
h zeta_K(1 + h) = (1 + o_{D}(1)) kappa_K,
uniformly for 0 < h <= 4 / log D,
and kappa_K^{-1} <<_n log D.
For the remaining finitely many fields (those with D below the
cutoff), the first comparison holds uniformly for sufficiently
small positive h, and kappa_K^{-1} <<_n 1.
Proof. Set L = log D.
STEP 1 (a clean zero-free region up to width 1/L).
Stark's Lemma 3 (Stark 1974): zeta_K has at most one zero in the
box Re(s) >= 1 - (4L)^{-1}, |Im(s)| <= (4L)^{-1}; such a zero, if
present, is real and simple. Stark's Lemma 8 (also applies to
non-normal K): a real zero beta >= 1 - (4 n! L)^{-1} forces K to
contain a quadratic subfield.
Since n is odd, K has no quadratic subfield (a quadratic subfield
would have degree 2 dividing n). So the hypothetical real zero
does not exist, and for a constant c_n > 0 every nontrivial zero
rho of zeta_K satisfies
|1 - rho| >= c_n / L.
STEP 2 (Hadamard identity and a bound at sigma_0 = 1 + 1/L).
Define S_K(sigma) = sum_rho Re(1/(sigma - rho)) for sigma > 1.
The classical Hadamard identity (Stark, eq. (9)) gives
S_K(sigma) =
1/sigma + 1/(sigma - 1) + L/2
+ zeta_K'/zeta_K(sigma)
+ G_n(sigma),
where G_n(sigma) = (n/2)(Gamma'(sigma/2)/Gamma(sigma/2) - log pi).
The zero-sum terms are nonnegative. G_n is bounded on [1, 2]. The
Euler product gives zeta_K' / zeta_K <= 0 on (1, infinity). At
sigma_0 = 1 + L^{-1},
S_K(sigma_0) <= 1/sigma_0 + L + 0 + 0 + O_n(1) <<_n L.
STEP 3 (persistence of the bound).
For sigma in (1, 1 + 4/L], rho = beta + i gamma, define
X = L (1 - beta), Y = L gamma, U = L (sigma - 1).
Then X >= 0, 0 < U <= 4, and |X + iY| >= c_n by STEP 1.
Compute:
Re( (sigma - rho)^{-1} ) / Re( (sigma_0 - rho)^{-1} )
= [(X + U)/(X + 1)] * [((X+1)^2 + Y^2) / ((X+U)^2 + Y^2)]
<= 4 * (1 + 3/c_n)^2.
The first factor is <= 4 because X >= 0 and U <= 4. The second
factor is bounded by a constant times (1 + 1/c_n)^2 via
|X + 1 + iY| <= |X + iY| + 1 <= (1 + 1/c_n) |X + iY| and
|X + U + iY| >= |X + iY|, giving the n-bounded factor shown.
Summing gives 0 <= S_K(sigma) <<_n S_K(sigma_0) <<_n L throughout
(1, 1 + 4/L].
STEP 4 (log derivative near the pole).
Define F_K(sigma) = (sigma - 1) zeta_K(sigma). Then
F_K' / F_K = 1/(sigma - 1) + zeta_K'/zeta_K(sigma)
= S_K(sigma) - 1/sigma - L/2 - G_n(sigma)
= O_n(L),
using the Hadamard identity re-solved for zeta_K'/zeta_K.
STEP 5 (F_K extends to sigma = 1).
Because zeta_K has a simple pole at s = 1 with residue kappa_K,
F_K extends analytically to sigma = 1 and F_K(1) = kappa_K > 0.
For 0 < h <= 4 / L:
|log F_K(1 + h) - log F_K(1)|
<= h * sup_{1 < sigma < 1+h} |F_K'/F_K|
<<_n h L <= 4 C_n.
So F_K(1 + h) / kappa_K is bounded above and below by constants
depending only on n. That is,
h zeta_K(1 + h) asymp_n kappa_K for 0 < h <= 4 / log D.
STEP 6 (residue lower bound).
At sigma_0 = 1 + L^{-1}, zeta_K(sigma_0) >= 1 (Euler product of
factors > 1 at sigma_0 > 1). So
F_K(sigma_0) = L^{-1} zeta_K(sigma_0) >= L^{-1}.
But F_K(sigma_0) = kappa_K (1 + O_n(1)) from STEP 5. So
kappa_K >>_n 1/L, hence kappa_K^{-1} <<_n log D.
STEP 7 (finiteness of the discarded fields).
Only finitely many fields of fixed degree n have discriminant
below a cutoff D_0. Direct proof: nonzero vectors of the
normalized lattice D^{-1/(2n)} sigma(R) have absolute coordinate
product at least D^{-1/2}; by Mahler's compactness criterion
this places the family {D^{-1/(2n)} sigma(R)} in a compact set
of the space of lattices. Local basis lifts and a finite cover
give bases of R with uniformly bounded conjugates. A basis
element outside Q generates K (n prime), and its monic integral
minimal polynomial therefore has bounded coefficients. There are
only finitely many such polynomials. Each corresponding zeta
function has a simple pole at 1; the remaining claims follow
from continuity and the fundamental identity at those finitely
many fields. QED.
LEMMA IV.2 (Shiu uniform).
Let b be a nonnegative multiplicative function with b(m) <=
d_n(m) for all m. Then for Q sufficiently large in terms of n,
sum_{u - v < m <= u} b(m)
<<_n (v / log u) * exp( sum_{l <= u} b(l) / l ),
for all u^{1/2} <= v <= u.
Proof. This is Pollack's Theorem 1.1 with all excluded residue
sets empty. To check uniformity:
d_n(l^j) = binom(j + n - 1, n - 1) <= n^j (j >= 1);
d_n(m) <<_{n, eps} m^{eps}.
Both are growth conditions required by Pollack. Pollack's strict
interval condition u^beta < v <= u is satisfied by taking
beta = 1/4; thus the theorem covers the stated range including
both endpoints. QED.
PROPOSITION IV.3 (weighted count).
Fix A_0 >= 1 and 0 < eta <= 1. For Q sufficiently large in
terms of n, A_0, eta,
sum_{x - y < N(a) <= x} P(a) <<_n kappa_K y / q,
for all x, y with Q^{3/4} <= x <= A_0 Q and eta x <= y <= x.
The implied constant is independent of A_0 and eta.
Proof.
STEP 1 (building the multiplicative comparison function).
Set b(m) = a_K(m) * 1_{(m, q) = 1}. This function is
nonnegative and multiplicative. It is bounded by d_n: at every
rational prime l (including ramified primes), the local
generating series for a_K is
prod_{p | l} (1 - T^{f_p})^{-1},
which is coefficientwise bounded by (1 - T)^{-n} because
sum_{p | l} f_p e_p = n and each e_p >= 1. So a_K(l^j) <= d_n(l^j)
for every j >= 0. Multiplicativity gives the global bound.
STEP 2 (the split).
Split an integral ideal a as a = a_S * a_cop, where
a_S = prod_{l in S} a_{(l)} has valuation-tuple k (= the
S-supported part), and a_cop has norm coprime to q. Because
P_l = 1 at every l not in S, the weight factorizes:
P(a) = P(k) * 1. So
sum_{x - y < N(a) <= x} P(a)
= sum_k P(k) sum_{(x-y)/d(k) < m <= x/d(k)} b(m).
(IV.4)
STEP 3 (Case 1: d(k) <= Q^{1/8}).
Put d = d(k) and u = x/d. Then u >= Q^{5/8}, so log u >=
(5/8) log Q and u >= D^{1/4}. Put h = 1/log u. Note h <=
4 / log D: since log u >= (5/8) log Q >= (5/8) log D and 1/log u
<= (8/5) / log D <= 4 / log D for log D >= 2. So Lemma IV.1
applies.
By Lemma IV.1: h zeta_K(1 + h) <<_n kappa_K, so
zeta_K(1 + h) <<_n kappa_K log u. Note: no upper restriction on
u. Increasing u only decreases h, so the lemma applies even when
D stays bounded and q grows.
Positivity of Euler logarithm:
sum_{l <= u} a_K(l) l^{-1-h} <= log zeta_K(1 + h).
So
exp( sum_{l <= u} a_K(l) l^{-1-h} ) <<_n kappa_K log u.
Since a_K(l) <= n on primes (n prime -> each rational prime
splits into at most n primes in K, each contributing at most 1
to a_K(l), and a_K(l) = 1 for unramified un-split primes, so
a_K(l) <= n for all l), and since sum_{l <= u} (log l) / l
<< log u, one has
sum_{l <= u} a_K(l) (l^{-1-h} - l^{-1}) <<_n h log u <= 1.
Consequently
exp( sum_{l <= u} a_K(l) / l ) <<_n kappa_K log u.
STEP 4 (remove S-primes).
One has log Z_S = sum_{l in S} log Z_l and, from the Euler
product,
log Z_l = sum_{l in S} a_K(l)/l + O_n(1).
So log Z_S = sum_{l in S, l <= u} a_K(l)/l + O_n(1) plus the tail
sum_{l in S, l > u} a_K(l)/l. Bound the tail: for those l,
a_K(l)/l <= n / l <= n / u.
The number of such l is at most (log q) / log u. So the tail is
<<_n log q / (u log u) * 1 <<_n log(2q) / Q^{5/8} <<_n 1.
Combining:
exp( sum_{l <= u} b(l) / l ) <<_n kappa_K log u / Z_S.
(IV.5)
STEP 5 (Shiu in the short interval).
The interval length v = y/d satisfies v >= eta u. Since u >= Q^{5/8},
for Q sufficiently large at fixed eta, eta u >= u^{1/2}.
Apply Shiu's theorem (Lemma IV.2) to the inner sum in (IV.4):
sum_{u - v < m <= u} b(m)
<<_n (v / log u) * exp( sum_{l <= u} b(l) / l )
<<_n (v / log u) * (kappa_K log u / Z_S)
= v kappa_K / Z_S.
Summing over k with d(k) <= Q^{1/8}, weighting by P(k):
contribution (Case 1)
<=_n (kappa_K / Z_S) sum_k P(k) (y / d(k))
= (kappa_K y / Z_S) sum_k P(k)/d(k)
<=_n (kappa_K y / Z_S) * (Z_S / q) [Lemma III.1]
= kappa_K y / q.
STEP 6 (Case 2: d(k) > Q^{1/8}).
Terms with d(k) > x vanish: the inner sum in (IV.4) is empty
because x/d < 1. For remaining terms x/d >= 1.
Elementary divisor-sum bound (valid for z >= 1):
sum_{m <= z} d_n(m) <= z (1 + log z)^{n-1}.
Reason: multiply out the n-fold divisor series and bound by
integrating the first n-1 variables.
Using the weight-tail (Lemma III.3) with s = 1/(2n):
contribution (Case 2)
<= C_n (x / q) * (1 + log x)^{n-1}
* sum_{d(k) > Q^{1/8}} P(k)/d(k)
<= C_n (x / q) * (1 + log x)^{n-1} * (q^{-1} Q^{-s/16})
= C_n (x / q) Q^{-s/16} (1 + log x)^{n-1}.
Bound: by Lemma IV.1 with q | log D, kappa_K^{-1} <<_n log(3D),
and D <= Q^2. Up to n-bounded constants, the ratio
(contribution Case 2) / (kappa_K x / q) --> 0
as Q -> infinity.
STEP 7 (closing).
At fixed A_0 and eta, increase the threshold for Q until the
Case 2 contribution is at most kappa_K eta x / q <= kappa_K y / q.
The Case 1 contribution is bounded by C_n kappa_K y / q with C_n
independent of A_0 and eta. So
sum_{x - y < N(a) <= x} P(a) <<_n kappa_K y / q.
The uniform-in-(A_0, eta) constants come from Lemma IV.1 (whose
implied constant only depends on n), the multiplicative divisor
bound, and Lemma III.1/III.2/III.3 (each of which has an
n-bounded implied constant). QED.
Tightness follows from a cube about the origin.
For 0 < theta < 1, define the open subset
Cup(theta) = { Lambda in X_n : exists 0 != v in Lambda,
||v||_infty < theta }.
A lattice in Cup(theta) contains a "short vector" relative to the
standard cube [-theta, theta]^n. Mahler's compactness criterion
(Mahler 1946) says: { Lambda : no nonzero vector with ||v|| < theta }
is compact in X_n. So the complement of Cup(theta) is compact, and
Cup(theta) exhausting X_n as theta -> 0 measures "escape to the
cusp."
PROPOSITION V.2 (tightness).
Every sequence of the packet probabilities with Q -> infinity is
tight. In particular, for 0 < theta < 1,
limsup_{Q -> infinity} mu_*
{ Lambda : exists 0 != v in Lambda, ||v||_infty < theta }
<= C_n theta^n.
Every subsequential probability limit is A_n-invariant.
Proof. Take f = 1_C where C = [-theta, theta]^n. Then E_f(Lambda)
counts nonzero v in Lambda cap C; on Cup(theta), E_f >= 1
because the short vector is in the open cube (still contained in
C, up to the boundary of measure zero, which does not matter for
integrals). So
mu_* ( Cup(theta) ) <= int_{X_n} E_f d mu_*.
Compute V_f(z) for this f.
For fixed z > 0, the C-orbit intersection with the cube is
parameterized by (u_1, ..., u_{n-1}) in R^{n-1} and one global
sign; the indicator of C is 1 iff
|omega_j e^{u_j}| < theta for j < n, and
|omega_n z e^{-u_1 - ... - u_{n-1}}| < theta.
The signs omega_j only affect the signs, not absolute values, so
the indicator depends only on the u_j. Set w_j = log theta - u_j
for j < n. Then
|omega_j e^{u_j}| < theta iff w_j > 0 (for j < n),
and the last condition becomes sum_{j<n} u_j > log(z/theta),
i.e., sum_{j<n} w_j < n log theta - log z = log(theta^n / z).
So the volume (in u-coordinates) of the set where the indicator
equals 1 is (for S = log(theta^n / z) > 0)
volume = S^{n-1} / (n-1)!.
Therefore
V_f(z) = 2^n / (n-1)! * max(0, log(theta^n / z))^{n-1}.
Set X = theta^n Q and U = Q^{3/4}. Applying Proposition II.7.1
(unfolding):
int E_f d mu_* =
(2^n / (n-1)!) * (1 / (sqrt(D) kappa_K))
* sum_{N(a) <= X} P(a) *
( max(0, log(X/N(a))) )^{n-1}.
So it remains to bound
S := sum_{N(a) <= X} P(a) (log(X/N(a)))^{n-1}.
Note that V_f vanishes for z > theta^n, i.e., N(a) > X = theta^n Q.
If X < 1, the sum is empty and the integral is zero.
BOUND A (small N(a) <= min(U, X)). One has
(log(X/N(a)))^{n-1} <= (log X)^{n-1}
because N(a) <= X means X/N(a) >= 1. Since X <= Q * 1 (theta < 1),
log X <= log Q. So
sum_{N(a) <= min(U, X)} P(a) (log(X/N(a)))^{n-1}
<= (log Q)^{n-1} sum_{N(a) <= U} P(a).
Apply Proposition IV.3 with x = y = U = Q^{3/4}:
sum_{N(a) <= U} P(a) << kappa_K U / q = kappa_K Q^{3/4}/q.
So the "small" contribution is
<<_n (log Q)^{n-1} kappa_K Q^{3/4}/q.
Divide by sqrt(D) kappa_K (in the unfolding formula):
<<_n (log Q)^{n-1} Q^{3/4} / (q sqrt(D))
= (log Q)^{n-1} Q^{-1/4}.
Since q sqrt(D) = Q, this is O_n(Q^{-1/4} (log Q)^{n-1}).
BOUND B (medium U < N(a) <= X). Define truncated cumulative
sum
A_U(t) = sum_{U < N(a) <= t} P(a), U <= t <= X.
Integration by parts:
sum_{U < N(a) <= X} P(a) (log(X/N(a)))^{n-1}
= (n-1) int_U^X A_U(t) (log(X/t))^{n-2} dt/t.
The boundary terms vanish because A_U(U) = 0 (empty sum over
interval of length zero) and (log(X/X))^{n-1} = 0.
By Proposition IV.3 at x = y = t (with t in [U, X], t >= Q^{3/4},
t <= X <= Q <= A_0 Q for A_0 = 1): A_U(t) << kappa_K t/q. So
S_medium <<_n (kappa_K/q) int_U^X (log(X/t))^{n-2} dt
= (kappa_K/q) X int_0^1 (log(1/v))^{n-2} dv
= (kappa_K/q) X * Gamma(n-1)
<<_n kappa_K X / q.
Divide by sqrt(D) kappa_K:
<<_n X/(q sqrt(D)) = theta^n Q / Q = theta^n.
Combine:
int E_f d mu_* <<_n theta^n + Q^{-1/4} (log Q)^{n-1}.
Letting Q -> infinity gives limsup mu_* (Cup(theta)) <= C_n theta^n.
TIGHTNESS. Fix a mass error eps > 0; choose theta small enough
that C_n theta^n < eps; then by the limsup bound there is Q_0 for
which every packet with Q >= Q_0 has mass <= eps on Cup(theta). The
complement of Cup(theta) is compact (Mahler). Finitely many packet
supports with Q < Q_0 are compact by the packet-model lemma. So the
whole sequence is tight.
INVARIANCE. Each packet measure is A_n-invariant by construction
(it is a weighted average of A_n-invariant orbit probabilities).
Any weak subsequential limit is therefore A_n-invariant, by
testing against continuous functions and using that A_n acts by
pushforward on the space of probabilities (pushforward of invariant
is invariant) and compactness of tight probability families. QED.
Small balls need a small-box argument.
For r > 0, define
B(r) = { h in SL_n(R) : max_{i, j} | h_{ij} - delta_{ij} | < r }.
This is the radius-r ball around the identity in the sup-norm on
matrices, intersected with SL_n(R). It is an open neighborhood of
the identity.
PROPOSITION VI.1 (ball bound).
Let mu be a weak limit of packet probabilities along Q -> infinity.
For every compact Omega subset X_n, there are C_Omega > 0 and
r_Omega > 0 such that
mu( x B(r) ) <= C_Omega r^n for all x in Omega, 0 < r < r_Omega.
Proof. The critical step is a "small-box" lower bound for V_f, then
feeding it into Proposition IV.3 gives the upper bound on
int E_f d mu.
STEP 1 (small boxes in a fixed compact of n-tuples).
Fix a compact V subset (R^*)^n and b_0 >= 1. For w in V and small
r, take
f = 1_{ w + [-b_0 r, b_0 r]^n }.
Each coordinate of w has fixed sign and is uniformly separated from
zero on V, and the box has logarithmic width O_V( r ) in every
coordinate. So the absolute coordinate product of every point of
the box lies within Cr of z_w = prod_j |w_j|, for a constant C
depending only on V, b_0, n. As a result,
V_f(z) <= C r^{n-1} * 1_[z_w - Cr, z_w + Cr](z).
Choose r_0 > 0 so small that z_w > 4C r_0 on V.
STEP 2 (apply weighted count).
Use Proposition IV.3 with
x = (z_w + 2Cr) Q, y = 4Cr Q.
Then x - y = (z_w - 2Cr) Q, and the slice height z_w avoids the
boundary. The V_f-support [z_w - Cr, z_w + Cr] Q is contained in
the interval (x - y, x].
Uniformity in w and Q: for small enough r <= r_0 and large enough
Q, the interval satisfies
Q^{3/4} <= x <= A_0 Q with A_0 = z_w + 2C r_0,
eta x <= y <= x with eta = 4Cr / A_0 <= 1.
(The constant A_0 grows with z_w, but z_w is bounded above on V.)
Apply Proposition IV.3:
sum_{x - y < N(a) <= x} P(a) <<_n kappa_K y / q.
Combine with V_f bound:
int E_f d mu_* <= (1 / (sqrt(D) kappa_K))
* C r^{n-1} * sum_{x-y < N(a) <= x} P(a)
<<_{V, b_0, n}
(1 / (sqrt(D) kappa_K)) * C r^{n-1} * (kappa_K y/q)
= (r^{n-1} / (sqrt(D))) * (C y / q)
= (C r^{n-1} / Q) * (y / 1) (using sqrt(D) q = Q)
= r^{n-1} * (C * 4CrQ) / Q
= 4C^2 r^n.
(Note: The last identity absorbs all constants into a single n-
bounded "C^2 r^n" because Q cancels.) The important structural
term is r^n, independent of Q.
STEP 3 (choosing w_x at each x in Omega).
A vector w_x must be chosen in each lattice x (for x in Omega)
with all coordinates nonzero, uniformly over x in Omega.
Claim: every full lattice has a vector with all nonzero
coordinates. Proof: take any Z-basis x_1, ..., x_n of the lattice;
the integer combination
x_1 + N x_2 + N^2 x_3 + ... + N^{n-1} x_n
has coordinates that are polynomials in N; each coordinate is not
identically zero, since the basis vectors are linearly independent.
Only finitely many integer N give a zero coordinate. Any other N
works.
Uniformize over Omega: near each center x in Omega, choose a
continuous local lift to basis matrices, and continue the chosen
vector using its fixed integer basis coefficients (the integer N
and the basis-change identification are locally constant on Omega,
so the chart data can be chosen uniformly). On a compact neighborhood inside
the chart, the coordinates of w_x stay bounded away from zero and
bounded above. A finite cover of Omega by such neighborhoods,
combined with a maximum over finitely many choices, gives choices
w_x in a fixed compact V' subset (R^*)^n.
This pointwise selection need not be continuous or measurable; it
is only used to control one test function per x, and no integration over x is made.
STEP 4 (reduce ball-mass to E_f at w_x).
Fix x in Omega. For h in B(r), each coordinate of w_x * h - w_x
satisfies
|(w_x h - w_x)_j| <= r * sum_{i} |(w_x)_i| <= b_0 r
for one fixed b_0 depending only on V', by Holder's inequality
applied entry-by-entry. So every lattice in x B(r) contains a
nonzero vector in w_x + [-b_0 r, b_0 r]^n, hence satisfies
E_f( x h ) >= 1 for h in B(r).
Consequently, for each fixed x in Omega and small fixed r,
mu_i( x B(r) ) <= int E_f d mu_i <<_{V', b_0, n} r^n.
The bound's implied constant is independent of r; the threshold
in Q_i may depend on r, which is exactly what is needed.
STEP 5 (pass to the limit mu).
Fix x and r. For all i large enough (threshold depending on r),
mu_i( x B(r) ) <= C_Omega r^n. Since x B(r) is open, Portmanteau
gives
mu( x B(r) ) <= liminf_i mu_i( x B(r) ) <= C_Omega r^n.
The same constant C_Omega works for every fixed x in Omega with
the same r-allowance. This proves the bound for all 0 < r < r_Omega
(the value r_Omega depending on the uniform r_0 in STEP 1). QED.
Positive entropy then closes the proof.
Choose real numbers lambda_1, ..., lambda_n, summing to zero and
distinct. Set
a(t) = diag(e^{lambda_1 t}, ..., e^{lambda_n t}), t in R,
lambda = min_{i != j} |lambda_i - lambda_j| > 0.
The entropy of the flow map x -> x a(1) is bounded from below.
For any A_n-invariant probability nu, write h_nu(a(1)) for the
measure-theoretic entropy of the transformation x -> x a(1)
with respect to nu.
PROPOSITION VII.1 (entropy from balls).
Let n >= 3 and let mu be an A_n-invariant probability on X_n.
Suppose for every compact Omega subset X_n there exist C_Omega,
r_Omega > 0 such that
mu( x B(r) ) <= C_Omega r^n for x in Omega, 0 < r < r_Omega.
Then every A_n-invariant probability xi satisfying xi <= c mu for
some finite c has h_xi( a(1) ) >= lambda / 3.
Proof. Fix 0 < b < 1/4 and consider the two-sided neighborhoods
T_t = a(t) B(b) a(-t) cap a(-t) B(b) a(t), t >= 0.
STEP 1 (structure of T_t).
If h in T_t, then
|h_ii - 1| < b, and |h_ij| < b e^{-|lambda_i - lambda_j| t}
for i != j.
(Standard computation: conjugation by a(t) multiplies the (i, j)
entry by e^{(lambda_i - lambda_j) t}, and the two-sided
intersection bounds both t and -t, so entries pick up the smaller
of the two decay factors, giving the absolute exponent.)
Set r = e^{-lambda t}. Every term in the determinant expansion of
det(h) other than the product of diagonal entries contains at least
two off-diagonal entries; each off-diagonal entry is bounded by
b r. There are n! - 1 such terms; the total of the terms is O_{n,b}(r^2).
Since det(h) = 1 (h in SL_n(R)), one gets
prod_i h_ii = 1 + O_{n, b}(r^2).
Consequently the diagonal matrix
d(h) = diag( h_11, ..., h_{n-1, n-1}, (prod_{i<n} h_ii)^{-1} )
lies in a fixed compact subset D subset A_n: its log entries are
bounded uniformly in t (and r -> 0), because all h_ii are needed
to be within b of 1 and the product correction is bounded.
Also, d(h)^{-1} h in B(C r) for a constant C = C(n, b), because
off-diagonal entries are O(r) and the diagonal correction is O(r^2)
plus the entrywise-close-to-1 diagonal.
STEP 2 (tube covering).
In log coordinates, D is a bounded subset of R^{n-1}. Cover D by a
mesh of size ~r (in log coords); there are at most N_t <= C r^{-(n-1)}
centers d_j. Enlarging C to absorb the matrix-multiplication error:
T_t subset union_{j=1}^{N_t} d_j B(C r). (VII.1)
STEP 3 (tube mass for xi).
Fix a compact Omega subset X_n. For x in Omega, all centers
x d_j lie in the fixed compact set Omega D. So domination xi <= c mu
and the ball bound give, for all sufficiently large t (i.e., small
r = e^{-lambda t}), for every x in Omega:
xi( x T_t )
<= sum_{j=1}^{N_t} xi( x d_j B(C r) ) [by (VII.1)]
<= N_t * c * C_Omega (Cr)^n [domination + ball bound]
<= C_{Omega, b, c} r^{-(n-1)} r^n
= C_{Omega, b, c} r
= C_{Omega, b, c} e^{-lambda t}. (VII.2)
Since xi is a probability, the constant can be enlarged (uniformly
in Omega) to cover the finitely many integer times t before the
estimate holds.
STEP 4 (apply ELMV's two-sided tube criterion).
The criterion is (Einsiedler-Lindenstrauss-Michel-Venkatesh, Duke 2009,
Corollary 3.3). In the notation
B^{(-s, t)} = a(-s) B a(s) cap a(-t) B a(t),
the criterion says the following. If flow-invariant probabilities
xi_i converge weakly to a probability xi, and t_i -> infinity, then
h_xi(a(1)) >= eta, provided that for every compact set Omega there
is an identity neighborhood B and a constant C_Omega such that
xi_i( x B^{(-t_i, t_i)} ) <= C_Omega e^{-2 eta t_i}
for all x in Omega.
The criterion applies on the noncompact lattice space; the
requirement "xi is a probability" is essential.
Apply with: constant sequence xi_i = xi, t_i = i, identity
neighborhood B = B(b), eta = lambda/3. The tube
B(b)^{(-t, t)} = T_t. From (VII.2),
xi( x T_t ) <= C e^{-lambda t}
<= C e^{-2 lambda t / 3} (since 1 >= 2/3)
= C e^{-2 eta t}.
So the criterion's hypothesis is met, and h_xi(a(1)) >= eta =
lambda/3. QED.
LEMMA VII.2 (entropy in ergodic components).
Under the hypotheses of Proposition VII.1, almost every A_n-ergodic
component of mu has positive entropy for a(1).
Proof. The group A_n = R^{n-1} is locally compact and second
countable, and its action on X_n (a standard Borel space) is
continuous. The ergodic decomposition theorem (Greschonig-Schmidt
2000, Theorem 5.2, specialized to invariant probabilities)
supplies a measurable decomposition
mu = int_Z nu_z d tau(z),
where each nu_z is an A_n-invariant and A_n-ergodic probability,
and Z is the quotient with respect to a factor fixed by A_n.
MEASURABILITY. Set T(x) = x a(1). The function z -> h_{nu_z}(T) is
measurable. Proof: choose increasing finite Borel partitions P_k
generating the Borel sigma-algebra of X_n. For every T-invariant
probability nu,
h_nu(T) = sup_k inf_{N >= 1} (1/N) H_nu( V_{j=0}^{N-1} T^{-j} P_k ).
H_nu denotes the Shannon entropy of a finite partition. The
right side is a countable (sup over k, inf over N, sum over j) of
Borel measurable functions of nu, so it is measurable in z.
ROKHLIN ENTROPY INTEGRAL. The component partition is fixed by T,
since a(1) in A_n. Rokhlin's entropy integral formula (Rokhlin
1967, Theorem 9.8) then gives
h_mu(T) = int_Z h_{nu_z}(T) d tau(z). (VII.3)
The same formula holds after restricting to any measurable
collection of components. It does not require the nu_z to be
T-ergodic: the induced transformation on the component space is
the identity and contributes zero entropy.
CONTRADICTION. Suppose Z_0 = { z : h_{nu_z}(T) = 0 } had weight
w = tau(Z_0) > 0. Then
xi = (1/w) int_{Z_0} nu_z d tau(z)
is an A_n-invariant probability satisfying xi <= w^{-1} mu.
Equation (VII.3), applied to the restricted decomposition,
gives h_xi(T) = 0, contradicting Proposition VII.1 (which
requires h_xi >= lambda/3 > 0). Thus almost every ergodic
component has positive entropy. QED.
By (EKL 2006, Theorem 1.3 and Corollary 1.4), for prime n >= 3,
any A_n-invariant A_n-ergodic probability on
SL_n(R) / SL_n(Z)
is the Haar measure m_n whenever some diagonal one-parameter
subgroup has positive entropy.
In the EKL convention the action is on the LEFT. The space X_n = SL_n(Z) \ SL_n(R) with
right action by a(t) matches EKL after the map
Gamma g -> g^{-1} Gamma.
Precisely: from a point Gamma g, right multiplication by a(t)
gives Gamma g a(t). Under inversion, that becomes
a(t)^{-1} g^{-1} Gamma = a(-t) g^{-1} Gamma, which is a LEFT
multiplication by a(-t). Entropy is unchanged by inversion of an
invertible transformation (and by replacing a one-parameter flow
by its inverse: entropy of a(t) equals entropy of a(-t)). So EKL
applies to the present convention with no additional hypothesis.
PROOF OF THE MAIN THEOREM.
Let mu_i be the packet probabilities along a sequence in the
theorem's hypothesis, so D(M_i) -> infinity.
(a) Q_i -> infinity. Follows from Claim II.1.1: D(M_i) <= Q_i^{2n}
and D(M_i) -> infinity forces Q_i -> infinity.
(b) Tightness. Proposition V.2 gives limsup mu_i(Cup(theta)) <=
C_n theta^n, so for small theta all the packet measures are
tight on the complement of the cusp.
(c) Invariance. Proposition V.2 also shows every subsequential
limit mu is A_n-invariant.
(d) Ball bound. Proposition VI.1 gives mu( x B(r) ) <= C_Omega r^n
for x in any compact Omega, uniformly on the limit mu.
(e) Positive entropy. Proposition VII.1 applies (hypotheses met
from (d)), giving h_xi(a(1)) >= lambda/3 for every
A_n-invariant xi <= c mu.
(f) Positive entropy in each ergodic component. Lemma VII.2
applies, so almost every A_n-ergodic component of mu has
positive entropy for a(1).
(g) Classification. Since n is prime and >= 3, the EKL
classification makes every A_n-invariant, A_n-ergodic,
positive-entropy probability on X_n equal to m_n.
(h) Reconstruction. Combine (f) and (g): every ergodic
component of mu equals m_n. Integrating the components
(ergodic decomposition) gives mu = m_n.
(i) Uniqueness and no mass loss. Every subsequential limit is
m_n, by (h). Together with tightness from (b), this proves
mu_i -> m_n as i -> infinity. And by (h) plus tightness,
no mass escapes into the cusp.
The estimates (b)-(e) were made for each prescribed local type
separately; no averaging over different local module types is used.
This closes the proof. QED.
This appendix identifies the packet measure with an adelic
pushforward of torus Haar probability. It is not needed for the
main proof; the purpose is to show that the lattice
parametrization derived above agrees with the standard
adelic formulation from the literature.
Let A_f denote the finite adeles of Q, and set
Z_hat = prod_l Z_l,
A_Q = R x A_f,
A_K = K (x)_Q A_Q,
R_hat = R (x)_Z Z_hat.
Choose once and for all a Q-basis b_1, ..., b_n of K and let
rho(alpha) denote multiplication by alpha on row coefficient
vectors in this basis. This embeds the algebraic torus
T_K = (Res_{K/Q} G_m) / G_m
into G = PGL_{n/Q}. The base matrix
B_sigma = ( sigma_j(b_i) )_{i, j}
has the useful property
rho(alpha) B_sigma = B_sigma diag(sigma_1(alpha), ..., sigma_n(alpha)).
This identity is called the regular diagonalization of rho(alpha).
It says that the rational multiplication matrix becomes a
diagonal matrix in the sigma-embedding, with the n conjugate
values along the diagonal.
Fix g_f in G(A_f). Choose any matrix lift to GL_n(A_f) (the
choice is unique modulo the center of GL_n, i.e., modulo
A_Q^*-values). Set
M_hat(g_f) = Z_hat^n g_f^{-1},
M(g_f) = K cap M_hat(g_f).
LEMMA VIII.1. For every g_f in G(A_f), the subset M(g_f) is a
full Z-lattice of K, and its pro-finite completion equals
M_hat(g_f).
Proof. Choose an integer d >= 1 with
d Z_hat^n subset M_hat(g_f) subset d^{-1} Z_hat^n.
(Existence: g_f is a product over primes of local matrices; only
finitely many are not integral, so d can bound their
denominators.) Patch the finite quotient d^{-1} Z^n / d Z^n: this
gives a full lattice M = K cap M_hat whose completion is
M_hat(g_f). Full rank: the Q-span of M is K because
M_hat(g_f) has full Q_hat-rank and K has dimension n.
Change of projective lift: if g_f is replaced by c g_f for a scalar
finite idele c in A_Q^*, then M_hat(g_f) is multiplied by c^{-1}.
Every such c is a rational scalar times an element of Z_hat^*; the
latter fixes Z_hat^n and hence every completed full lattice.
So the change of lift rescales M(g_f) by a rational number,
which corresponds to the same K^*-homothety class. QED.
Fix g_inf in G(R) with
g_inf^{-1} T_K(R)^circ g_inf = A_n.
T_K(R)^circ is the identity component of the real points of
T_K. Existence: the real points of T_K are a maximal R-split
torus in G(R) = PGL_n(R); its identity component is a conjugate
of A_n inside PGL_n(R). The odd-degree identification
G(R) = SL_n(R) (via [h] -> (det h)^{-1/n} h) transports this to
A_n subset SL_n(R).
Define the adelic packet as the projection of
T_K(Q) \ T_K(A_Q) g
to G(Q) \ G(A_Q) / G(Z_hat), equipped with the pushforward of
torus Haar probability. Set g = (g_inf, g_f) in G(A_Q), and
G(Z_hat) is the standard compact-open subgroup.
PROPOSITION VIII.2 (adelic-packet).
Assume the data above. Then:
(i) The adelic torus quotient T_K(Q) \ T_K(A_Q) is compact.
(ii) The pushforward of torus Haar is a probability on
G(Q) \ G(A_Q) / G(Z_hat); after the standard identification
with X_n, it agrees with mu_{K, M(g_f), sigma'} for some
ordering sigma' of the real embeddings of K.
(iii) Every prescribed local homothety type occurs this way as
g_f varies over G(A_f).
(iv) Changing the rational basis (b_i) or the projective matrix
lift of g has no effect on the packet.
(v) Changing the allowed real diagonalizer (g_inf) only reorders
the real embeddings sigma' but does not change the packet.
(vi) Therefore the main theorem includes every adelic
presentation of any local homothety type.
Proof. The identifications are worked through step by step.
STEP 1 (ambient quotient ~ X_n). Given a matrix lift g_f in
GL_n(A_f), choose a basis matrix c in GL_n(Q) for the Q-span of
Q^n cap Z_hat^n g_f^{-1}. Then c g_f in GL_n(Z_hat), so the
number-field side produces the real lattice Z^n (c g_inf),
which after normalization to covolume one lies in X_n. Changing
the basis c or the lift g_f does not change that lattice.
For odd n, the signed real nth root [h] -> (det h)^{-1/n} h
identifies G(R) = PGL_n(R) with SL_n(R), and G(Z) = PGL_n(Z)
with SL_n(Z). So the setting is X_n = SL_n(Z) \ SL_n(R).
STEP 2 (torus quotient).
Multiplication matrices modulo scalars give
T_K(Q) \ T_K(A_Q) ~ K^* \ A_K^* / A_Q^*.
To see the inclusion direction: a matrix whose projective class
lies in T_K becomes, over any ground field F_0 containing Q, a
scalar multiple of a multiplication operator over the algebraic
closure of F_0. Multiplication operators commute with the regular
representation of K, so the matrix belongs to (K (x) F_0)^*.
The centralizer of the regular representation is precisely the
algebra of multiplication operators, so the projective class is
K^* x F_0^*. This works already over F_0 = Q. Finite lifts may be
chosen integral at almost all primes, giving the compact open
finite part.
STEP 3 (finite idele action).
A finite idele t_f in A_f^* acts on the completed lattice by
M_hat |--> t_f^{-1} M_hat.
Thus the finite-idele translates of a fixed completed lattice are
exactly the local homothety type prescribed (the
choice t_f ranges over the finite ideles, whose action is
precisely by local multipliers c_l in K_l^*).
STEP 4 (stabilizer).
The stabilizer of M_hat(g_f) in A_K^* is
{ z in A_K^* : z M_hat = M_hat } = (O(M) (x) Z_hat)^*.
Equality holds precisely when both z and z^{-1} preserve M_hat.
Modulo the central A_Q^* (which corresponds to the PGL reduction),
the projective stabilizer is the image of (O(M) (x) Z_hat)^* in
T_K(A). This unit group is compact open in the finite ideles and
has finite index in R_hat^*; at each prime l not dividing q, it
equals R_l^*. Its projective image is compact open as well. (No
invertibility of M over O(M) is used: the stabilizer condition is
additive and only depends on M_hat as a lattice.)
STEP 5 (decomposition, measures, matching to the packet model).
Divide T_K(A) by the image of R_hat^*. Fractional-ideal
factorization together with
A_Q^* = Q^* x R^+ x Z_hat^*
gives, for each ordinary ideal class [I], one real fiber
E \ C (E = R^*, C as in the packet model). These fibers are compact by
the unit theorem. The omitted finite unit group is compact, so
the whole torus quotient is compact. This proves (i).
At the stabilizer of M_hat, the finite fibers retain the
translates N(I) from the packet model. The same decomposition becomes
disjoint union over [I] of ( N(I) x C ) / E,
with exactly the action from the packet-model lemma (Lemma
II.5.1). Torus Haar measure pushes to:
- uniform counting measure on each finite unit orbit N(I) (this
is the "label" part);
- the half-log-Haar measure on C, i.e., counting on 2^n sign
components times Lebesgue on the u-coordinates;
- after normalization, the packet probability mu_{K, M(g_f),
sigma'} from the packet model.
For g_inf = [B_sigma], the regular diagonalization identifies
the resulting real lattices with the displayed map (Q N(I))^{-1/n}
sigma(N) t of Lemma II.5.1. This proves (ii).
STEP 6 (occurrence of every local type, and independence).
Every full lattice M in K can be obtained by choosing a basis
matrix for its completed lattice and taking its inverse as g_f.
So (iii) holds: every local type is realized by some g_f.
For (iv): if a rational basis change matrix is c in GL_n(Q), then
B'_sigma = c B_sigma and rho' = c rho c^{-1}, while the packet data
have representatives g'_infty = c g_infty and g'_f = c g_f. The
torus is conjugated and the translated packet is left-multiplied
by c, which disappears in the ambient rational quotient
G(Q) \ G(A_Q) / G(Z_hat). So the packet is unchanged. Also, any
other allowed real diagonalizer must be of the form [B_sigma d P]
with d real diagonal and P a permutation matrix, because it must
permute the joint eigenspaces. The diagonal factor translates the
real torus and permutes its sign components, preserving its Haar
probability; the permutation reorders the embeddings. So the
packet is unchanged up to re-indexing sigma. This proves (v).
Combining (ii)-(v), the main theorem includes every adelic
presentation of any local homothety type. QED.