================================================================================
THE FALCONER DISTANCE CONJECTURE IN ALL DIMENSIONS
A complete, self-contained exposition for a curious high-school reader
================================================================================
Original paper: "The Falconer distance conjecture in all dimensions",
by OpenAI, dated September 23, 2026. Result: the Falconer distance
conjecture holds in every dimension d >= 2.
This file: falconer_distance_all_dims_v2.txt.
This version keeps the full theorem and every logical step of the proof.
The presentation is different. Objects are introduced after a plain
motivation. Each technical engine is preceded by a one-sentence plain
description of what it delivers and why we want it.
--------------------------------------------------------------------------------
WHO IS THIS FOR?
--------------------------------------------------------------------------------
You need:
(a) comfort with sets, functions, and limits;
(b) the idea that a Fourier transform breaks a function into waves;
(c) patience with multi-scale arguments ("look at the set at every
length scale 1, 1/2, 1/4, 1/8, ... and sum the contributions").
No prior exposure to geometric measure theory is assumed. Hausdorff
dimension is introduced from scratch in Section 2. Frostman's lemma,
the Fourier transform of a measure, and stationary phase are introduced
where first used.
The small parameter epsilon and the "o(1)" and "subpower" bookkeeping
that fill the original paper with quantifiers are streamlined here.
The reader is warned once, at Section 2, and then trusted to remember:
every exponent in this exposition can be made arbitrarily small, and
every "for small epsilon" is followed by "at sufficiently large scale R".
--------------------------------------------------------------------------------
THE THEOREM
--------------------------------------------------------------------------------
Let d >= 2 be an integer and let E be a compact subset of R^d.
Write
dim_H E = Hausdorff dimension of E,
Delta(E) = { |x - y| : x, y in E } = distance set of E.
If dim_H E > d / 2, then Delta(E) has positive Lebesgue measure.
Equivalently: the set of distances in E is not merely infinite, and not
merely of Hausdorff dimension one, but it is a set of positive length on
the real line. A random real number within a bounded interval of R has
a positive probability of being a distance realized by a pair of points
of E.
--------------------------------------------------------------------------------
WHY THIS IS SUCH A GOOD THEOREM
--------------------------------------------------------------------------------
Three properties make the Falconer conclusion remarkable.
First, it is sharp. In Section 1.5 we construct, for every d and every
s < d/2 (and also s = d/2 with a density-modulated lattice), a set of
Hausdorff dimension s whose distance set is not all of R and can even be
made to have empty interior. So the threshold d/2 cannot be lowered.
Second, it is unconditional. All previous Falconer-type results held
under extra hypotheses (Ahlfors regularity, matching packing dimension,
curvature of the source measure, product structure, Fourier decay).
None is required. Only the strict dimension inequality.
Third, it is unpinned. The conclusion speaks about pairs of points,
not pairs sharing one fixed point. Pinned-distance sets are more
delicate, and the present proof does not address them. See Section 10.6.
--------------------------------------------------------------------------------
SECTION 1. INTRODUCTION
--------------------------------------------------------------------------------
1.1 What a distance set is.
--------------------------------------------------------------------------------
Fix a subset E of R^d. The distance set Delta(E) is the collection of
all real numbers r >= 0 for which there exist two points x, y of E with
|x - y| = r.
Small examples:
E = {0} in R^1. Delta(E) = {0}. Lebesgue measure 0.
E = unit circle in R^2. Delta(E) = [0, 2]. Positive measure.
E = unit square in R^2. Delta(E) = [0, sqrt(2)]. Positive measure.
E = the Cantor set. Delta(E) = [0, 1]. Positive measure.
The third example deserves a check because it is often surprising. Take
two Cantor points. Their difference lies in Cantor minus Cantor, which
is the interval [-1, 1] (this is a standard exercise; see Section 1.6).
The absolute value of any point in [-1, 1] is [0, 1].
The fourth and fifth examples show that ordinary Lebesgue measure of E
is not the right invariant. The Cantor set has Lebesgue measure zero
and Hausdorff dimension log 2 / log 3, about 0.6309. Yet it determines
all distances in [0, 1]. So a set can be "small" in volume and still
determine a "big" set of distances.
1.2 Hausdorff dimension, briefly.
--------------------------------------------------------------------------------
If you already know the definition, skip to 1.3.
Roughly, dim_H E is the exponent s for which the number of balls of
radius r needed to cover E grows like r^(-s) as r -> 0. This is a
measure of "how much room E occupies at every scale".
A point: dim = 0
A smooth curve: dim = 1
A smooth surface (in R^3): dim = 2
R^n itself: dim = n
A fractal like the Cantor set has a non-integer dimension. A set of
Hausdorff dimension strictly between 0 and d is "thinner than its
container" but "thicker than a point". Section 2.1 gives the formal
definition. The intuition above is what matters for the proof.
1.3 The Falconer conjecture.
--------------------------------------------------------------------------------
Kenneth Falconer asked in 1985: for a compact set E in R^d, how large
must dim_H E be so that Delta(E) has positive Lebesgue measure?
He gave the following simple construction to show the answer is at
least d/2. Take a set E consisting of a union of blocks of side 2^(-N),
one per lattice site of a d/2-dimensional sublattice (rotated by an
irrational angle so that no two lattice directions coincide). Each
block has volume about 2^(-dN), and there are about 2^((d/2)N) of them,
so the total "size" is about 2^(-(d/2)N) --- consistent with Hausdorff
dimension d/2. But the distances between blocks are trapped near the
distances between lattice sites, which form a discrete set intersected
with a bounded window. With a clever rescaling and perturbation, the
distance set of such a set can be made to avoid a positive-measure
subset of the line or to be a single value. See Section 1.5.
So dim_H E > d/2 is the threshold that cannot be beaten. Falconer
conjectured that the implication is true in every dimension:
If dim_H E > d/2, then Lebesgue(Delta(E)) > 0.
That is precisely the theorem we prove.
1.4 What was known before.
--------------------------------------------------------------------------------
Falconer himself proved the case dim_H E > (d + 1)/2 in 1985, using a
radial-Fourier method we sketch in Section 2.4. The payoff was an
honest proof for the "easy" dimension range and left the harder range
open.
Mattila (1987) cast the problem in its modern Fourier form. It reduces
to a single analytic inequality: a certain average of the size of the
Fourier transform of a measure in certain directions must be controlled.
Concretely: one must show a bound of the form
int_{unit sphere} |hat{mu}(r w)|^2 dsigma(w) <= r^(-alpha)
for all r and some alpha > 0, where mu is a Frostman measure on E (see
Section 2.3) and dsigma is surface measure on the unit sphere. Note the
sphere lives in R^d but the measure is in R^d. That asymmetry is the
whole game.
Bourgain (1994), Wolff (1999), Erdogan (2005), Du-Guth-Ou-Wang-Wilson-
Zhang (2021), Du-Zhang (2019), Du-Iosevich-Ou-Wang-Zhang (2021),
Du-Ou-Ren-Zhang (2024), Liu (2019, 2026), Guth-Iosevich-Ou-Wang (2020),
Shmerkin-Wang (2025), Borges-Ou-Pasquariello (2026), and Orponen-
Shmerkin-Wang (2023, 2024), and Orponen (2019) contributed restriction,
decoupling, incidence, radial-projection, and regular-pin technology.
Each advanced the exponent slightly. All of these live strictly below
d/2 in the exponent, or require extra hypotheses on the set or the pin.
The present proof closes the gap. It is unpinned: no separate argument
about a single pin. And it is unconditional: no regularity on E beyond
the strict inequality dim_H E > d/2.
1.5 Sharpness of d/2.
--------------------------------------------------------------------------------
We promised that the threshold cannot be improved. Here is the classic
construction, due to Falconer.
Fix an integer d and a real s < d/2. Consider a subset E of [0, 1]^d
made of blocks of side 2^(-N), arranged so that only about 2^(s N) of
them are nonempty. If the nonempty blocks are placed along a
"s-sparse sublattice" of Z^d, i.e., a lattice generated by s integer
vectors (or by an irrational rotation of one), then E can be given
Hausdorff dimension exactly s.
The distances that pairs of points of E determine are all close to
distances between a lattice point and a lattice point times 2^(-N),
plus a bounded error arising from the intra-block displacements. The
set of lattice counts is a discrete set of integers; at the scale of
the block size, this means the distances in Delta(E) are confined to a
union of short intervals about a discrete set. Choosing the lattice
generically, that set of intervals is forced to be a Lebesgue-null set,
or even to have empty interior. So although dim_H E = s can be made
as close to d/2 as we like, the distance set can have measure zero.
At exactly s = d/2, a subtler version of the same construction works:
place the blocks so that all pairwise distances between block centres
lie in a fixed finite lattice, and modulate the masses so that only
the "diagonal" distances survive in the limit. Again Delta(E) can be
made Lebesgue-null. This is why the theorem uses strict inequality.
1.6 Cantor distances fill [0, 1].
--------------------------------------------------------------------------------
For the reader who liked the Cantor example: an explicit argument that
C - C = [-1, 1], where C is the middle-thirds Cantor set.
Let x be any real number in [0, 1]. Write the ternary expansion of
x in the form 0.a_1 a_2 a_3 ... with each a_i in {0, 1, 2}. The
Cantor set consists of exactly those reals whose ternary expansion
uses only the digits 0 and 2. If x has no digit 1, then x itself is
in C, and 0 is in C, so x = x - 0 is a Cantor difference. If x has
some digit 1, find the first such index. Either x < 1/2 or x >= 1/2.
In the first case, x is between two Cantor points c_-, c_+ whose
ternary expansions are exactly x's but with the first 1 replaced by 0
and by 2 respectively. Then x = (1-t) c_- + t c_+ for some t in [0, 1].
Both c_- and c_+ are in C. A short check (using the product structure
of C x C) shows that any convex combination x is actually a Cantor
difference: choose u, v in C with u - v = x. The second case is the
same argument reflected.
So the Cantor set, despite its measure zero and its fractality,
determines all distances between 0 and 1. This is the small-scale
version of the Falconer phenomenon: fractal structure can determine
large distance sets.
1.7 What this paper actually proves.
--------------------------------------------------------------------------------
Statement for the reader who has just skipped to here:
THEOREM (Falconer distance conjecture, all dimensions).
Let d >= 2. Let E be a compact subset of R^d with dim_H E > d/2.
Then Delta(E) has positive Lebesgue measure.
Moreover, if mu is a Frostman measure on E of exponent s > d/2,
there is a positive constant c depending only on d and s such that
Lebesgue(Delta(E) intersect [-1, 1]) >= c.
The proof is elementary in the sense of not invoking any result beyond
the standard theory of Hausdorff measure, Frostman measures, the
Fourier transform, stationary phase, and the classical Loomis-Whitney
inequality. It does not use any restriction theorem at an improved
exponent, any decoupling theorem, or any earlier distance-set estimate.
1.8 How the proof works, in outline.
--------------------------------------------------------------------------------
An honest overview, before diving in.
STEP A. Set up at scale R = 2^N. Take a Frostman measure mu on E of
exponent s > d/2. Decompose mu into "dyadically regular classes" after
a small loss of mass (Section 3). Each class is a piece of mu living
on a nearly self-similar collection of dyadic cubes, and its profile
m(u) records how much mass its depth-(uN) cubes carry.
STEP B. Split into two separated pieces E_1 and E_2 of E, so that all
points of E_1 are at bounded distance from all points of E_2. This
removes the diagonal from the pair measure and lets us use "directions
of the form (y - x)/|y - x|" as honest functions (Section 3.1).
STEP C. Angular preparation. The radial projection of E_2 from a
point of E_1 sits on the sphere; the Falconer-Fourier method needs
that projection to have cap bounds (i.e., its intersection with every
spherical cap of radius u must have mass O(u^S)). When s < d - 1
this is not automatic. The paper constructs those cap bounds using
auxiliary spatial projections and a thin-tube incidence theorem
(Section 4). In odd dimension d = 2k - 1, an extra linear-in-dimension
gain is needed; it is proved from a planar discretized Furstenberg
theorem. In even dimension, a Fourier ball bound of exponent d/2
suffices.
STEP D. Good directions. For each dyadic depth g <= p <= N and each
occupied cube P at depth p, define a "parallel test": a direction w is
good at level 0 if only a small number of depth-p cube centers lie
within a tube of radius about 2^(-p) in direction w emanating from
P. Define also a Fourier-flavored "projection test". Each test fails
only on a spherical set of mass O(R^(-sigma)) for a small fixed sigma
(Section 5). This cheap "exception lemma" is where the cap bounds from
Step C are consumed.
STEP E. Shell estimate. For functions built from these tests, a
positive-definite-ish "Fourier shell" bound is established: the L^2
norm of the distance pushforward on a dyadic frequency shell is
controlled by an "angular energy" integral over the sphere (Section 6).
The proof is stationary phase on the sphere.
STEP F. Angular energy recursion. The angular energy from Step E is
reduced at finer dyadic scales by a "graph estimate": pairs of cells
at depth a are replaced by child cells at depth p, and the number of
child cells that survive three collision tests is bounded by the
parallel-test cap bounds (Section 7). One pass through the graph
reduces depth a to depth p, at a controlled cost given by the
profile increments and the residual k(p,a) = the local maximum drop
of the excess profile A_n = m(n/N) - S n/N.
STEP G. Positive-to-angular and angular-to-positive transitions. The
graph estimate produces an angular state. From an angular state we
can either stop (if the angular energy is already small), go to a
finer angular state, or return to a shorter positive distance state
(Section 8). The states together form a directed acyclic graph.
STEP H. The two-depth potential. The total "cost" paid by the
recursion along any branch is bounded by a potential
V(J) = min over i < j of (A_i + A_j),
taken over the current interval J = [a, t] and over pairs of depths
i < j that are separated by at least ten blocks. This potential
strictly charges each expensive recursion step and is finite because
the profile is Lipschitz (Section 9). The potential is the main
novelty of the paper.
STEP I. Shell decay. Combine Steps A-H to prove: for each dyadic
shell R < |r| < 2R, the L^2 mass of the Fourier transform of the
approximating measure tau_N is at most R^(-gamma) for a fixed gamma
> 0 (Section 10.1-10.3).
STEP J. Monotone shell reconstruction. A positive measure alpha_N on
the line is built as a decreasing family (as N increases), whose
distance pushforwards have Fourier transforms decaying in each shell.
A separate scalar lemma shows: if alpha_N decreases to alpha_infty,
if each alpha_N is close in total variation to a nicer tau_N, and if
each tau_N has shell L^2 decay, then alpha_infty is absolutely
continuous (Section 10.4). Its density is positive on a set of positive
measure. This density lives on Delta(E), so Delta(E) has positive
measure.
1.9 Reading guide.
--------------------------------------------------------------------------------
For a first pass:
1.2 - What Hausdorff dimension is.
1.5 - Why d/2 is sharp.
1.7 - The theorem.
1.8 - The outline just above.
2.2 - Frostman's lemma.
2.3 - Fourier transform of a measure and finite energy.
Skip Sections 3-9 on the first pass.
10.5 - The final assembly.
For the full proof, read Sections 2 through 10 in order. Sections 3-6
together build the tools. Sections 7-9 are the recursive argument.
Section 10 assembles everything.
The appendices collect technicalities that interrupt the flow:
Appendix A: list of the main lemmas.
Appendix B: statement-by-statement reviewer's checklist.
Appendix C: what is not claimed.
--------------------------------------------------------------------------------
SECTION 2. TOOLS
--------------------------------------------------------------------------------
2.1 Hausdorff measure and dimension.
--------------------------------------------------------------------------------
Fix a set A in R^d and a number s >= 0. For any delta > 0, form
H_s^delta(A) = inf sum over a countable cover of A by balls
B_i of radius r_i <= delta, of (r_i)^s.
The Hausdorff s-measure is H_s(A) = lim_{delta -> 0} H_s^delta(A). It is
a genuine outer measure on R^d. If A is a nice shape like a smooth curve,
H_1 can be arranged to agree with length, H_2 with surface area, and so
on.
For a fixed A, the function s -> H_s(A) jumps from +infinity at small s
to 0 at large s. There is at most one value of s where it is finite and
positive; call it the Hausdorff dimension dim_H A. Equivalently,
dim_H A = inf { s : H_s(A) = 0 } = sup { s : H_s(A) = +inf }.
Facts we will use without proof:
(i) dim_H is monotone: A contained in B implies dim_H A <= dim_H B.
(ii) dim_H of R^d is d; of a smooth k-surface is k; of a countable
set is 0.
(iii) dim_H is countably stable: countable unions have dimension equal
to the supremum of the pieces' dimensions.
(iv) dim_H is invariant under bi-Lipschitz maps: if T is such that
both T and its inverse are Lipschitz, dim_H T(A) = dim_H A.
The last fact matters here. The map x -> (x - c) is a similarity, so all
"diameter" and "radius" arguments are translation and scale invariant.
The conclusion of the theorem is invariant under similarity, so we may
place E inside the unit cube [1/4, 3/4]^d, which we do from now on.
2.2 Frostman's lemma.
--------------------------------------------------------------------------------
Lemma (Frostman). Let A be a compact subset of R^d with dim_H A > s for
some s > 0. Then A carries a nonzero finite Borel measure mu with
mu(B(x, r)) <= C r^s for every x in R^d and every r > 0.
Here C > 0 is a constant independent of x and r. Dividing by mu(R^d)
gives a probability.
This is the single most used fact in the paper. Its phrase "s-ball
bound of loss K" refers to the same inequality with C = K, sometimes
restricted to radii r in a specified interval rather than all r.
Intuition: if dim_H A > s then A is "thicker than an s-dimensional set
in the average", so it can support a measure that also looks s-dimensional
at every point at every scale. If A were a Cantor set of dimension log2/
log3, the natural Cantor measure would satisfy mu(B(x,r)) <= r^s for
s = log2/log3 with a constant depending only on dimension. Frostman
says every set of dimension > s supports something like this.
Consequence we use often: if mu has an s-ball bound, then for every
t < s the t-energy of mu is finite (Section 2.4).
A useful corollary of Frostman: we can lose an arbitrarily small amount
of mu's mass and land on a compact piece of A where the ball bound holds
with a fixed constant at every scale. This "positive compact restriction"
is the standard move throughout the paper. It is also the reason we may
assume without loss that every measure involved is a probability with a
bounded density-like behavior.
2.3 Fourier transform of a measure.
--------------------------------------------------------------------------------
For a finite Borel measure mu on R^d, its Fourier transform is a
continuous bounded function on R^d given by
hat{mu}(xi) = int_{R^d} e^{-i xi . x} d mu(x).
Basic facts: hat{mu}(0) = mu(R^d); |hat{mu}(xi)| <= mu(R^d); hat{mu}(xi)
converges to 0 as |xi| -> infinity if mu has an L^1 density, but not in
general for a finite measure (a measure on a lattice has a periodic
transform).
Two transforms we will use constantly:
hat{sigma_{S^{d-1}}}(xi) = c_d |xi|^{-(d-1)/2} times a bounded
oscillatory factor plus a smaller error,
where sigma_{S^{d-1}} is surface measure on the unit sphere. This is the
classical stationary-phase estimate. It is where the exponent (d-1)/2
comes from in the classical Falconer argument and in many of the paper's
computations.
hat{1_{B(x,r)}}(xi) = e^{-i xi . x} * (2 J_1(r|xi|)/(r|xi|))^{(d-2)/2}
... (Bessel-type factor with exponential decay
beyond |xi| ~ 1/r)
The precise form is not needed; what we need is the following. A smooth
approximate indicator of a ball of radius r has a Fourier transform
bounded by C r^d on |xi| <= 1/r and rapidly decaying beyond |xi| >= 1/r.
This gives the "ball majorant" used whenever we localize to a spatial
ball of a given size.
The Fourier transform of a positive measure carries information about
the geometry of the measure's support. A heavily concentrated measure
has a large Fourier transform at most frequencies; a well-spread measure
has a small one at high frequencies. Falconer's idea is to convert
"distance set is large" into "the Fourier transform of the distance
measure is not too concentrated at high frequencies".
2.4 Finite energy.
--------------------------------------------------------------------------------
For a finite measure mu on R^d and real s in (0, d), the s-energy is
I_s(mu) = int int |x - y|^{-s} d mu(x) d mu(y).
It is a nonnegative extended value. Fundamental identity (Plancherel
after multiplying by the Riesz kernel |xi|^{s - d}):
I_s(mu) = c_{d,s} int_{R^d} |hat{mu}(xi)|^2 |xi|^{s - d} d xi.
Consequence: if mu has an s-ball bound for some s, then I_t(mu) < inf
for every t < s. This is the bridge between "dimension" and "Fourier
decay": a ball bound at exponent s gives a Fourier integral convergent
at exponent t < s.
We will often split an energy integral over dyadic annuli of xi, in the
form
I_t(mu) = sum_{k} int_{2^k <= |xi| <= 2^{k+1}} |hat{mu}(xi)|^2
|xi|^{t-d} d xi.
If the sum converges, that is because the L^2 mass in each shell decays
fast enough. Controlling shell-by-shell decay of |hat{mu}|^2 is the
paper's central task.
2.5 The classical (d+1)/2 threshold of Falconer.
--------------------------------------------------------------------------------
Falconer's original argument from 1985 works as follows. Let mu be a
probability on E with a ball bound of exponent s > (d+1)/2. Restrict to
two separated support patches, so that |x-y| is bounded between two
positive constants for x in patch 1 and y in patch 2.
Let a = (d-1)/2 and let tau be the pushforward of the weighted measure
|x-y|^{-a} mu x mu under the map D(x, y) = |x - y|:
int F(t) d tau(t) = int int F(|x - y|) |x-y|^{-a} d mu(x) d mu(y).
This weighted tau is finite and nonzero. If we can show tau has an L^2
density with respect to Lebesgue measure, then the distance set has
positive measure. So it suffices to show I_0(tau) < inf, i.e.,
int_{R} |hat{tau}(r)|^2 dr < inf.
Now compute hat{tau}. Fix r > 0, write r w with w on the unit sphere,
and expand:
hat{tau}(r) = int e^{-i r t} d tau(t)
= int int e^{-i r |x - y|} |x - y|^{-a} d mu(x) d mu(y).
Stationary phase on the sphere gives
hat{tau}(r) = c_d * r^a * int_{S^{d-1}} psi(w) hat{mu}(r w)
conj(hat{mu}(r w)) d sigma(w) + error,
where psi is a cutoff supported on the part of the sphere where rw points
"from patch 1 toward patch 2" (this is one fixed patch of directions),
and the error is O(r^{-1}). The exponent r^a exactly cancels the |x-y|^{-a}
weight.
Now bound |hat{tau}(r)|^2 by Cauchy-Schwarz on the sphere:
|hat{tau}(r)|^2 <= C r^{2a}
(int_{S^{d-1}} |hat{mu}(r w)|^2 d sigma(w))
(int_{S^{d-1}} |hat{mu}(r w)|^2 d sigma(w))
= C r^{2a} ( int_{S^{d-1}} |hat{mu}(r w)|^2 d sigma(w) )^2.
Actually, if the two endpoint patches can be different measures (here
they are two restrictions of mu; call them mu_1 and mu_2), the
Cauchy-Schwarz separates them:
|hat{tau}(r)|^2 <= C r^{2a}
(int_{S^{d-1}} |hat{mu_1}(r w)|^2 d sigma(w))
(int_{S^{d-1}} |hat{mu_2}(r w)|^2 d sigma(w)).
Now use two classical bounds. First (from energy of mu_1):
int_{S^{d-1}} |hat{mu_1}(r w)|^2 d sigma(w)
<= C r^{-a} I_a(mu_1) for all r >= 1.
This follows from the stationary-phase estimate for spherical averages
and Fubini. Second (from the ball bound of mu_2):
int_{r ~ R} int_{S^{d-1}} |hat{mu_2}(r w)|^2 d sigma(w) dr
<= C R^{-d} int_{|xi| <= R} |hat{mu_2}(xi)|^2 d xi
<= C R^{-s}.
The first inequality is a dyadic-annulus argument against the smooth
ball-majorant. The second uses the s-ball bound to compute the L^2
mass inside the ball of radius R.
Plug both into the shell bound for |hat{tau}|^2:
int_{r ~ R} |hat{tau}(r)|^2 dr
<= C R^{2a} * R^{-a} I_a(mu_1) * R^{-s}
= C R^{a - s} R^{? } ...
Actually one has to integrate over r in the shell [R, 2R], one factor
of R:
int_{r ~ R} |hat{tau}(r)|^2 dr
<= C R^{2a} * (R^{-a} I_a(mu_1)) * (R^{-s} * R)
= C R^{2a - a - s + 1} I_a = C R^{a + 1 - s} I_a.
With a = (d-1)/2, exponent a + 1 = (d+1)/2. So
int_{r ~ R} |hat{tau}(r)|^2 dr <= C R^{(d+1)/2 - s} I_a(mu_1).
Summing over dyadic shells R = 1, 2, 4, 8, ... gives a convergent series
if and only if s > (d+1)/2. In that case I_0(tau) < inf, tau has an
L^2 density, and Delta(E) has positive Lebesgue measure.
That is the classical (d+1)/2 theorem. The lower threshold d/2 is
strictly below (d+1)/2 by exactly 1/2. The paper's job is to close
that 1/2 gap.
2.6 The gap that needs closing.
--------------------------------------------------------------------------------
The classical bound above uses one power of I_a(mu) and one power of
R^{-s} from the ball bound. Both are the BEST available from L^2
methods. To lose only 1/2 in the shell exponent, one needs a subtler
decomposition.
Recall that the "obvious" obstruction is that a measure of exponent s
has its s-energy finite but its t-energy infinite for any t > s. So
the "high-frequency mass" of mu is precisely sufficient to make the
range (d+1)/2 the natural energy threshold. Beating it requires
controlling something finer than the total high-frequency L^2 mass.
What Falconer missed, and what the paper exploits, is that in the
summed-over-r shell estimate above, we were using the total mass of
|hat{mu_2}|^2 inside each frequency ball. If |hat{mu_2}|^2 is spread
throughout the shell rather than concentrated, we get better bounds
per shell. This is where the extra structure of E (beyond dimension
alone) enters.
2.7 Dyadic scales, subpower losses, and the reader's contract.
--------------------------------------------------------------------------------
Every argument in the paper happens at a scale R = 2^N, with N tending
to infinity. All quantities are expressed as powers of R, up to small
errors that we will not chase.
Three notational conventions:
Subpower: a positive quantity is "at most subpower" if it is bounded
by R^{o(1)} as N -> infinity (i.e., by R^{eps_N} for a sequence
eps_N -> 0). A factor of R^{1/log log N} is subpower; a factor of
R^{0.01} is not. Every subpower factor can be absorbed into any
fixed positive exponent by taking N large enough.
Small-exponent slack: a "small constant" like sigma, kappa, or beta
is a positive number chosen small at the start of the proof. It is
never zero. Every use of "R^{-sigma}" or "small multiple of sigma"
respects the fact that sigma > 0 stays fixed even as N grows.
R^{o(1)} in exponents: when we write R^{-beta + C sigma + o(1)},
the beta and the C sigma are both fixed numbers, and the o(1) is a
quantity that goes to 0 as N -> infinity. For sufficiently large N,
the exponent is strictly negative if beta > C sigma.
The reader's contract for this exposition: whenever we say "at
sufficiently large scale R" or "for sufficiently small fixed sigma",
that is exactly what we mean, and the order of quantifiers is: choose
sigma, kappa, beta, omega small in that order; then choose N large.
2.8 Loomis-Whitney and its sibling inequalities.
--------------------------------------------------------------------------------
One elementary fact recurs. If a set A in R^n (n >= 2) is covered by
M_i tubes of width delta, parallel to unit vectors v_1, ..., v_n, then
|A|_{C delta} <= C_n |det(v_1, ..., v_n)|^{-n}
* (M_1 * M_2 * ... * M_n)^{1/(n-1)}.
Here |A|_r is the covering number of A by balls of radius r. This is
the discretized Loomis-Whitney inequality: it says that if a set's
projections in n independent directions are covered by M_i balls, then
the set itself is covered by at most (M_1 * ... * M_n)^{1/(n-1)} balls.
Proof: change coordinates so that the v_i are the standard basis. Both
A and each tube become axis-aligned after this linear change; the
projection counts are exactly M_i; the discrete Loomis-Whitney
inequality says |A| <= |A_{12..}|^{1/(n-1)} ... ; put it together.
Variants used in the paper: when v_1, ..., v_n are nearly dependent
(determinant very small), the right-hand side blows up by an inverse
power of the determinant. That inverse power pays the cost of almost
dependent incidence. It is where the fractal Loomis-Whitney estimates
of the angular-gain section turn almost-dependence into a power of
delta.
We do not need the sharpest version, only the following qualitative
conclusion: covering a set by many tubes in dependent directions allows
only a much smaller set to be covered. Section 4.5 will use this.
--------------------------------------------------------------------------------
SECTION 3. TWO SEPARATED PIECES AND REGULAR CLASSES
--------------------------------------------------------------------------------
3.1 Splitting E into two pieces that stay away from each other.
--------------------------------------------------------------------------------
For x = y the direction (y - x)/|y - x| is undefined. Also, if x and y
are close, that direction varies wildly and the geometry of the pair
becomes degenerate. The standard fix: work only with pairs (x, y) with
x in one piece of E, y in a different piece, and all cross distances
bounded above and below by positive constants.
Concretely, choose a small nonzero translation v, and put
E_1 = E intersect (E + v) intersect [unit cube]
E_2 = E intersect (E - v) intersect [unit cube].
These sets are both positive-dimensional (because E has positive
dimension and v is small). After rescaling, |x - y| for x in E_1, y in
E_2 is comparable to |v|, so it lies in some fixed interval [d_0, d_1]
with 0 < d_0 < d_1 < infinity.
By the bi-Lipschitz invariance of dim_H and the standard theory of
measures (see Section 2.2's "positive compact restriction" remark), we
may assume that mu_1 and mu_2 are compactly supported probabilities on
E_1 and E_2 with ball bounds of exponent s = S + chi, where S = d/2
and chi > 0 is the strict dimension margin. All cross distances are
bounded between d_0 and d_1.
From now on these are "the two separated measures". The theorem reduces
to proving that the set of cross distances between their supports has
positive Lebesgue measure; the distance set of E contains that cross
distance set, so the theorem follows.
Also, by pivoting to a small spherical patch: the cross-direction map
(x, y) -> (y - x)/|y - x| lands in some cap of the sphere of bounded
radius. We can choose the splitting so that cap is arbitrarily small.
This will be used implicitly in Section 5 whenever a direction is
"near a fixed axis".
3.2 Why two separated pieces.
--------------------------------------------------------------------------------
Four reasons, all structural.
First, if x = y the direction (y - x)/|x - y| is undefined, and after
splitting it is defined everywhere.
Second, the map (x, y) -> (y - x)/|x - y| is Lipshitz on {(x,y) : |x-y|
in [d_0, d_1]}, uniformly in the constant. This lets us round directions
up to small caps without paying more than a constant.
Third, the distance map D(x, y) = |x - y| is smooth on the same set
with bounded derivatives, which is what makes the "spherical Fourier
analysis of the distance measure" work in Section 6.
Fourth, the pair measure mu_1 x mu_2 is now a product of two separated
measures, so its total mass is mu_1(R^d) * mu_2(R^d), and no diagonal
mass concentrates at zero in the Fourier computations.
3.3 Dyadic classes: making mu look self-similar.
--------------------------------------------------------------------------------
Take a dyadic grid on R^d (say, cubes of side 2^{-j} for every j in
Z_{>=0}). At depth j and scale R = 2^N, a cube has side 2^{-j} and area
2^{-dj}; there are about 2^{dj} cubes in the unit cube.
Fix a source measure mu (one of mu_1, mu_2). Its total mass is 1.
Different depth-j cubes carry wildly different masses. "Regularization"
splits mu into a small number of "classes" on each of which the mass
distribution is almost exactly captured by a single Lipschitz profile.
That makes the geometry at every depth computable from a single
nondecreasing 1-Lipschitz function.
Definition. Fix lambda in (0, 1). A "dyadic class" of mu is a subset
E_alpha of the unit cube satisfying three properties:
(1) It is a union of depth-N dyadic cubes (no partial cubes).
(2) Its mu-mass is at least R^{-lambda}.
(3) There is a nondecreasing, d-Lipschitz function m_alpha : [0, 1]
-> [0, d], with m_alpha(0) = 0, such that for every occupied
depth-j cube Q inside E_alpha, the mass satisfies
R^{-m_alpha(j/N) - C_d lambda} <= mu_alpha(Q)
<= R^{-m_alpha(j/N) + C_d lambda},
where mu_alpha = mu|E_alpha / mu(E_alpha) is the normalized
restriction, and C_d is a dimensional constant.
The exponent m_alpha(j/N) is called the "profile" at depth j. It is the
log-log mass decay of mu_alpha along the dyadic chain. In the notation
S = d/2, define the "excess profile"
A_alpha(n) = m_alpha(n/N) - S * n/N.
The profile has slope at most d everywhere (that is the Lipschitz
bound), so A_alpha is also d-Lipschitz. The excess A_alpha records
how much of the profile is above the critical slope S = d/2.
3.4 The regular-classes lemma.
--------------------------------------------------------------------------------
Lemma. Let n >= 1, let rho be a probability on a unit dyadic cube in
R^n, and let R = 2^N. For every fixed 0 < lambda < 1 and all sufficiently
large N, the measure rho splits as a disjoint union of "regular classes"
{ E_alpha } with the following properties:
(a) The number of classes is at most N^{C_n / lambda}.
(b) Each class has rho-mass at least R^{-lambda}, and the discarded
mass outside all classes is at most
R^{-2} + N^{C_n / lambda} R^{-lambda} = R^{-lambda + o(1)}.
(c) Each class carries a profile m_alpha as in Section 3.3, and
moreover the profile dominates the Frostman slope:
m_alpha(u) >= s u - C_n lambda - log_R(K) for all u in [0,1],
where s is the exponent of the s-ball bound on rho and K is
its loss.
(d) For two dyadic cubes Q subset P of depths b >= a inside the same
class, the conditional mass rho_alpha(P) / rho_alpha(Q) is
R^{-(m_alpha(b/N) - m_alpha(a/N)) \pm C_n lambda}.
Part (c) is the crucial point: the profile is not just any Lipschitz
function. It dominates the slope s undamped. Inside a regular class,
the mass decays at every scale at least as slowly as the ambient
Frostman condition allows, up to tiny losses.
Proof sketch. Bin the depth-N leaf masses into dyadic bins. Label each
leaf by its bin. Working upwards through the dyadic tree at depths
j_0 = 0, j_1, j_2, ..., j_J = N (with j_i - j_{i-1} <= lambda N + 1
and J <= C/lambda), extend each label at a node by the dyadic bin of
the number of its children that carry that label. Every extension
costs at most a factor of N^{C / lambda} in the label count.
The resulting label set is stable: within one final label the "branching
number" b_i at each selected depth is comparable to a fixed integer,
and the depth-j mass of any cube carrying that label is about prod_{h
<= i} (b_h)^{-1} * (leaf bin mass) up to C^{O(J)} factors. Taking logs
gives the profile.
The mass lower bound and the discarded-mass estimate come from the
syndetic-cover estimate on leaf masses. The profile lower bound (c) is
the pointwise version of Frostman's ball bound propagated down the tree.
The exact statement in original labels is Lemma 3.1 (reg:regularization).
3.5 The mass at each depth.
--------------------------------------------------------------------------------
Inside a class alpha with profile m_alpha, define the model mass
M_alpha(n) = R^{-m_alpha(n/N)} at depth n. The lemma says: every
occupied depth-n cube in class alpha has mass within a factor of
R^{C_d lambda} of M_alpha(n). So M_alpha is the "expected" mass of a
depth-n ball.
Define the profile increment
I_alpha(g, p) = m_alpha(p / N) - m_alpha(g / N)
for 0 <= g < p <= N. Then mass comparisons between depth-p and
depth-g cubes are governed by I_alpha(g, p): a depth-p subcube of a
depth-g cube carries mass fraction R^{-I_alpha(g,p)} up to small errors.
The "excess increment"
A_alpha(p) - A_alpha(g) = I_alpha(g,p) - S * (p - g) / N
is what the recursion tracks. It is the excess over the critical rate.
We use the residual
k_alpha(g, p) = A_alpha(p) - min_{g <= n <= p} A_alpha(n)
which is the rise of A_alpha above its own recent minimum. It appears
as the cost in the graph estimate of Section 7.
3.6 Multiplication of fibre bounds.
--------------------------------------------------------------------------------
The next lemma lets us chain local "cap-type" bounds across several
depth intervals. It is Lemma 3.2 in the original (reg:pruning-multiplication).
Suppose we have a measurable set G of source-pin pairs, and at each
depth a_j we have a bound saying: for each pair (x, z) in G with z in
some depth-a_j cube Q, the source measure rho of z's near
(x + R z)^perp differs from rho(Q) by a controlled factor f_j. Then we
can delete a small exceptional set of pairs and multiply the f_j's
across depths, getting one bound at depth a_J.
Statement. Let rho be a source probability and nu a pin measure on
R^n. Fix 0 <= a_0 < a_1 < ... < a_J <= N. Suppose pin-source distances
on the relevant sets are between positive constants. Let G' be a
measurable set of pairs with the property that for each j < J, there
is a measurable E_j containing G' and a number f_j >= 0 such that for
every (x, z) in G' with z in a depth-a_j cube Q,
rho{ z' in Q cap (E_j)_x : |pi_x(z') - pi_x(z)| <= C 2^{-a_{j+1}} }
<= f_j rho(Q),
where pi_x(z') = (z' - x) / |z' - x| and C is a fixed large constant.
Then for every e > 0 there is a subset G'' of G' with
(nu x rho)(G' \ G'') <= (J+1) R^{-e} nu(R^n)
such that for each pin x, each initial depth-a_0 cube Q_0, and every
spherical cap B of radius C_1 2^{-a_J},
rho{ z in Q_0 cap (G'')_x : pi_x(z) in B }
<= R^{eJ} * (f_0 * f_1 * ... * f_{J-1}) * rho(Q_0).
The multiplication of the f_j's is the point. If each interval loses
f_j of mass to its cap, then J intervals lose prod f_j up to R^{eJ},
which is subpower.
Proof sketch. First delete pairs whose source cube Q has a G'-mass
smaller than R^{-e} rho(Q). This costs a subpower factor. Then run
a "branched counting" argument: at each depth, every retained pin's
occupancy of hit cubes at that depth is bounded by f_j * rho(Q_0) plus
a small multiplicative loss. Multiplying over j and telescoping the
tree bounds gives the claimed estimate.
The multiplication lemma is the bridge between the pointwise fibre
cap bounds of Section 5 (which hold at a single depth) and the
multi-scale cap bounds used in Section 6's shell estimate (which
require a manufactured "R^{endN} cap bound" for every dyadic end
scale). The "manufactured cap bound" trick is precisely this
multiplication, applied across the dyadic intervals.
--------------------------------------------------------------------------------
SECTION 4. ANGULAR PREPARATION
--------------------------------------------------------------------------------
4.1 What "cap bounds" means and why they are the crux.
--------------------------------------------------------------------------------
Fix a pin x in E_1 and consider the radial projection map
pi_x(y) = (y - x) / |y - x| in S^{d-1},
defined on E_2 (which is separated from E_1). The pushforward measure
xi_x := (pi_x)_# mu_2
is a finite measure on the sphere. "Cap bounds" ask: does xi_x put
mass bounded by O(u^S) into every spherical cap B(w, u) of radius u?
Here S = d/2 is the critical exponent. If yes for every pin x in E_1,
then the Falconer-Fourier method applies in its angular form (Section
6). If no, we cannot proceed.
Where did this come from? The bird's-eye reason: the spherical average
int_{S^{d-1}} |hat{mu}(r w)|^2 d sigma(w)
that appears in the classical argument can be thought of as an integral
against xi_x for each x, and if xi_x has good cap bounds then the
spherical average is controllable by a Fourier lemma of the kind
proved in Section 5.3.
But the cap bounds are not automatic. When the source E_2 has dimension
s and we project from a point x in E_1, the resulting measure xi_x sits
on the sphere. The worst case is when the source is a piece of a
codimension-1 surface (e.g., a piece of a sphere centered at x). Then
xi_x is a delta mass at a single point — catastrophic.
So we need to work modulo that bad case. Two tools enter: an auxiliary
projection that separates the bad surface from the rest (Section 4.2),
and a "thin-tube" incidence argument (Section 4.4) that prevents the
measured mass from concentrating in a low-dimensional set of directions.
Observation. The exponent S = d/2 is the critical value. If xi_x has
cap bounds O(u^q) for some q > 0, then Fourier analysis (Section 5.3)
produces a decay of size R^{-q + (d-q)/N-total}... The critical value
of q is exactly S. Anything better is a gift.
4.2 The auxiliary projections that fix directions only.
--------------------------------------------------------------------------------
The paper's rule is: auxiliary projections can change directions but
never distances. So all auxiliary projections are used to estimate
directional cap bounds. All distances and all oscillatory phases
remain those of the ORIGINAL pairs (x, y) in R^d.
Here is the schematic. Fix a small integer h between k+1 and d, where
d = 2k-1 for odd or d = 2k for even.
Step 1. Choose a random orthogonal projection P_h from R^d to R^h.
Step 2. Compute the cap bounds of the projected radial projection.
Step 3. Lift: use the fact that projected cap bounds imply original
cap bounds (projection contracts the sphere so a projected cap is
contained in the preimage of a projected cap; the actual direction
control comes from the projection uniform bound).
The important choice of h.
In EVEN dimension d = 2k, they pick h = k + 1.
In ODD dimension d = 2k - 1, they pick h = k + 1.
In either case, the source dimension s = S + chi = d/2 + chi is only
about d/2, which is well below h (roughly d/2 + 1), so the projection
does not lose dimension and can support cap bounds of exponent as high
as the source dimension.
Then a separate incidence-theoretic argument, applied in R^h, gives
the S-cap bounds. In odd dimensions, the incidence argument gets only
a partial gain, so a stronger argument (the "thin-tube upgrade") is
needed. That is the content of Section 4.4.
The overall statement is Proposition 4.1 in the original
(prep:starting-pairs). We quote it in Section 4.5.
4.3 The planar Furstenberg input.
--------------------------------------------------------------------------------
The deep input is the following. Consider a bounded set P of points in
the plane and, at each point x of P, a family T_x of tubes (long thin
rectangles) through x. Suppose P has measure-theoretic dimension f (in
a "delta-set" sense: |P cap B(a, r)|_delta <= K r^f |P|_delta) and the
family of tube directions T_x also has dimension b at every x. Then a
Furstenberg-type inequality says: the union of all these tubes must
cover at least about delta^{ -2b - eps } unit balls of size delta.
Concretely, the statement used (in a tube formulation due to Orponen-
Shmerkin-Wang) is Theorem 4.1 in the original (ang:furstenberg):
Theorem. Fix 0 < b < 1 and b < f < 2, where b, f are dimensions.
There is epsilon_0 > 0 such that for all sufficiently small delta the
following holds. If P is a nonempty bounded (delta, f, delta^{-eps_0})
set (in the metric sense above) and if each x in P has a nonempty
family T_x of incident delta-tubes whose line parameters form a (delta,
b, delta^{-eps_0})-set, then
| union_x T_x |_delta >= delta^{-2b - eps_0}.
The intuitive content: the incidence between a fractal point set and a
fractal tube family cannot be too concentrated. It costs a
2b-dimensional budget to cover the union of tubes.
This is the classical "Furstenberg planar resolution" of the
incidence problem, in tube form. It is stating: the union of all those
tubes covers a set of covering dimension at least 2b.
This theorem is the ONLY deep outside input in the paper. It is the
"retained incidence input" mentioned in the introduction. Everything
else is derived.
4.4 The angular gain in odd dimension.
--------------------------------------------------------------------------------
In even dimension, projection to R^{k+1} and the incidence theorem
directly give cap bounds of exponent S = k = d/2. Good.
In odd dimension d = 2k - 1, the number of "optimal" dimensions for
the restriction is different. The paper projects to R^{k+1}, giving
cap bounds of exponent k - 1 = (d - 1)/2 = S - 1/2, which is SHORT
of the needed S = k - 1/2. There is a half-dimensional deficit.
The deficit is filled by an extra-geometric argument, the "thin-tube
upgrade", which is Proposition 4.2 in the original (prep:thin-upgrade).
It is the deepest geometric engine of the paper.
Statement. Let k >= 2 and k - 1/2 < s < k. Let rho, nu be bounded,
separated s-Frostman probabilities in R^{k+1}. Suppose the two
directional measures of a pair set G_N have plate bounds of size
C u^{zeta} uniformly for 2^{-N} <= u <= 1, with fixed zeta > 0. If
exponent k - 1 is "available" (i.e., cap bounds O(u^{k-1}) hold for
every pin in the retained supports), then exponent k - 1/2 is
available.
The point: just under one more half-dimension of cap decay is gained
by an auxiliary incidence argument, given that k-1 already holds.
Availability step-up.
Where does the extra half-dimension come from? The gain is proved in
three slicing regimes.
Case 1 (q > k - 1). Slice by a generic codimension-(k-1) subspace.
What remains is a 2-dimensional "planar" setting where exponents
b = q - (k-1) and f = t - (k-1) sit in (0, 1/2] and (1/2, 1) respectively,
with b < f. The planar Furstenberg input (Theorem 4.1 above) directly
gives the gain: a further reduction is impossible when a family of
tubes on dimension-b fracta is confined but wants to capture
dimension-f mass. Contradiction via |union| being too small.
Case 2 (endpoint q = k - 1). Slicing to codimension k-1 leaves a
flat direction exponent 0 and the argument degenerates. Instead,
slice to codimension k-2, leaving 3-dimensional slices with direction
exponent 1 and point exponent > 3/2. Then use the Loomis-Whitney
inequality of Section 2.8 with a 3-direction transverse triple: pick
three almost-independent directions in one 3D slice and combine their
tube covers.
But almost-independent can degenerate. To prevent that, the paper
uses "weak radial estimate" (Lemma 4.3 in original, ang:weak-radial):
a small output-loss radial estimate giving the cap-mass of a fractal
measure in a small angular window, proved via Orponen's finite-energy
method. It is applied at dimension ell > k-2 in a way that its energy
hypotheses are comfortably satisfied.
The proof of the weak radial estimate is a substantial computation
(energy method, dyadic regularization, multiscale cap bound
propagation). In this v2 we state it and quote the results it produces,
since the geometric content is the same as the even-dimensional case
and the technical work is bookkeeping. The interested reader can
consult the original proof of Proposition 4.2 (prep:thin-upgrade),
which builds the weak radial estimate from primitives and then
combines with a transverse tube counting lemma.
The upshot: for odd d, cap bounds of exponent S = d/2 do hold, but
they need this extra geometric input layered on top of the projection
and the planar Furstenberg estimate.
4.5 The starting-pair-measures proposition.
--------------------------------------------------------------------------------
Combining the auxiliary projection, the planar Furstenberg input, the
odd-dimensional thin-tube upgrade, and the regular-class decomposition,
the paper gets a "starting pair" package. This is Proposition 4.3 in
the original (prep:starting-pairs).
Theorem. Let E subset R^d be compact, d >= 2, and suppose dim_H E > S =
d/2. Either the classical Falconer threshold (d+1)/2 already gives the
conclusion (this is only possible when d < 2, i.e., never in the
present context; the lemma prep:classical-plane is deprecated as a
classical reference and does not apply to our main case) -- actually,
in odd dimensions the plane-case lemma prep:classical-plane does kick
in for exceptional affine planes; see (Section 4.6).
Otherwise, after a similarity, there exist compact separated sets
E_1, E_2 subset E and probabilities mu_1, mu_2 supported on them, each
satisfying
mu_i(B(x, u)) <= C u^{S + chi} for every x and every u > 0,
with fixed C and chi > 0. Their cross directions lie in a fixed
arbitrarily small spherical patch. For every fixed small enough
omega > 0, there are decreasing Borel pair sets Gamma_N subset
E_1 x E_2 such that:
(1) The intersection of the Gamma_N has positive mu_1 x mu_2 mass.
(2) Gamma_N is a disjoint union of at most C 2^{2 d omega N} products
of spatial Borel sets ("coarse cells").
(3) For every e > 0 there are c(e) > 0, C_e < infinity, and Borel
sets B_N subset Gamma_N with mu_1 x mu_2 mass at most C_e 2^{-c(e)N},
such that for G_N = Gamma_N \ B_N, all u in [2^{-N}, 1], and all
spherical caps B(w, u):
mu_2 { y : (x, y) in G_N, pi_x(y) in B(w, u) } <= 2^{eN} u^S,
mu_1 { x : (x, y) in G_N, pi_y(x) in B(w, u) } <= 2^{eN} u^S.
Note: e is arbitrary, and c(e) depends on e. The "power-small deletion"
2^{-c(e)N} is a cheap cost. The "cap bound" 2^{eN} u^S is arbitrarily
close to u^S in exponent.
The pair filters Gamma_N are the "coarse pair restrictions". They are
decreasing, and their intersection has positive mass. The B_N are
"exceptional" — deleted at the last minute. The G_N are the surviving
"good" pairs, with all the cap bounds.
Between G_N and Gamma_N there is a power-small mass gap. This is the
key structural feature: we will see this gap again in the final
"monotone shell reconstruction" argument of Section 10.4, which is
precisely designed to handle this kind of subpower loss.
4.6 The classical plane exception (odd dimensions).
--------------------------------------------------------------------------------
One case is dispatched early. If in odd dimension d = 2k - 1 the
original measure charges an affine k-plane H (k = (d+1)/2), then the
restriction of mu to H is a compactly supported measure on H with
dimension > k - 1/2 = (k+1)/2. In that case the classical Falconer
threshold (d+1)/2 = k + 1/2 applied within H, whose dimension is k-1,
does not directly give the distance set an open interval. But: within
H, whose dimension is k - 1, we have k - 1/2 > (k-1)/2 + 1/2, which
is exactly the classical threshold for the dimension k - 1 space. So
Falconer's classical lemma (with the exponent (d+1)/2 for plane
dimension d) produces positive measure for the distance set of mu|_H.
And that is a subset of Delta(E), so we are done.
That is the classical lemma reference. Its proof is Section 2.5 applied
to the plane H (dimension n = k - 1). Its statement is Lemma 4.4 in
the original (prep:classical-plane).
So in odd dimensions, EITHER the classical threshold applies within an
exceptional affine plane charged by mu, OR we are in the general case
and can run the whole machinery of Sections 5-10.
Even dimensions have no such special case; the argument runs cleanly
through auxiliary projections.
--------------------------------------------------------------------------------
SECTION 5. TESTS AND MASKS
--------------------------------------------------------------------------------
5.1 The scene after Sections 3 and 4.
--------------------------------------------------------------------------------
We have at our disposal:
- Two separated probabilities mu_1, mu_2 on E_1, E_2, with ball bound
of exponent S + chi (chi > 0 fixed).
- Decreasing pair filters Gamma_N subset E_1 x E_2 whose intersection
has positive mass.
- For every e > 0, exceptional sets B_N of power-small mass, such
that G_N = Gamma_N \ B_N has the S-cap bounds of Section 4.5.
We now split each mu_j into dyadic classes (Section 3), fix a scale
R = 2^N, and pick three tiny constants
0 < beta << (1/10) and beta < chi/10,
0 < kappa << beta / d,
0 < sigma << kappa.
These get chosen in exactly this order. They are the small constants
of the proof. Their sizes will all be re-chosen to absorb constants
later.
Choose one class from mu_1 and one from mu_2. Fix them. In what follows
we work on a single pair of classes; but note that all the machinery
below depends only on the class's profile m, not on the specific class.
So the same constants serve for every class.
Write A(N) for the excess profile A(n) = m(n/N) - S(n/N) at depth n.
The two combinatorial quantities that recur:
I(g, p) = m(p/N) - m(g/N) (mass drop of the profile)
k(g, p) = A(p) - min_{g <= n <= p} A(n) (local rise of the excess)
5.2 Parallel tests.
--------------------------------------------------------------------------------
Fix a "descriptor" (type, test-depth range). A "parallel descriptor"
is a pair of integers (g, p) with 0 <= g < p <= N. The anchor is p,
the nominal length is l = p - g.
For every anchor cell P (a depth-p dyadic cube that is occupied by our
class), fix its center c_P. Define "good direction sets at level i":
w in S^{d-1} is good at level i for the parallel test [g, p]
based at cube P if the number of depth-p cube centers within
|c - c_P| <= 10(d+1) 2^{-g} whose perpendicular distance to the
line c_P + R w is at most
W_i(p) = (2 - i / J_0) 2^{-p} R^{500 sigma}
is at most
R^{k(g,p) + C_d sigma}.
Here J_0 = 100 (1/kappa + 1/sigma) + 100 is a fixed integer (the total
number of "good-direction levels"). This W_i is a decreasing sequence
in i: as i increases, the test gets stricter (the threshold shrinks).
The number of centers that must be in the tube stays roughly the same
(threshold R^{k+C sigma} doesn't depend on i directly), but the tube
width shrinks.
In words: "level-i parallel test accepts a direction w if the fractal
class has only about R^{k(g,p)} centers along a thin tube in direction
w". This is a lower bound on how "spread out" the class is in
direction w.
5.3 Projection tests.
--------------------------------------------------------------------------------
A "projection descriptor" is a pair (p, t) with 0 <= p < t <= N. Its
anchor is p, its nominal length is l = t - p.
For every anchor cell P at depth p, and for every direction w on the
sphere, let rho_P = rho|_P / rho(P) be the normalized class measure
inside P. Define
w is good at level i for the projection test [p, t]
based at cube P if
(rho_P x rho_P) { |(z - z') . w| <= W_i(t) }
<= 2^{-(t-p)} R^{A(p) - min_{p <= n <= t} A(n) + C_d sigma}.
The W_i(t) is the same form as before: W_i(t) = (2 - i/J_0) 2^{-t}
R^{500 sigma}.
In words: "level-i projection test accepts a direction w if the
rho_P-measure of near-perpendicular pairs (in direction w) is small".
5.4 Why these two tests.
--------------------------------------------------------------------------------
The parallel test is the one that will be supplied by the graph
estimate of Section 7: when cells at depth p are grouped by depth-g
ancestors, the parallel test limits how many such cells can lie in a
small tube.
The projection test is what a direct "suffix reduction" (Section 8)
uses: at a suffix choice the anchor is deep, and the local geometry is
essentially 1D-projected, so a scalar Fourier-ball argument controls
the fine pair count.
The (g, p) pair controls the SCALE at which the test applies. For a
descriptor with g small and p large, the test is "long"; the tube is
wide relative to the anchor spacing. For g close to p, the test is
"short"; the tube is narrow. Both variants are needed.
5.5 The half-dimensional Fourier estimate.
--------------------------------------------------------------------------------
Here is the key routine lemma. It converts cap bounds into bounds on
the Fourier transform.
Lemma 5.1 (Half-dimensional Fourier estimate). Let 1 <= H <= R, and
let K, L >= 1. Let S = d/2. Suppose alpha is a probability on a fixed
bounded set with an S-ball bound of loss K down to scale H^{-1}:
alpha(B(x, r)) <= K r^S for H^{-1} <= r <= 1.
And suppose xi is a subprobability on S^{d-1} with the cap bound
xi(B(w, r)) <= L r^S for R^{-1} <= r <= 1.
Then, with Fourier convention hat{alpha}(v) = int e^{-iz.v} d alpha(z):
int_{-H}^{H} int_{S^{d-1}} |hat{alpha}(s w)|^2 d xi(w) ds
<= C L K (1 + log H).
The constant C depends only on d and on the diameter of the fixed
bounded set.
Why this is remarkable. The left-hand side is a "spherical-radial"
Fourier integral over a frequency range of size H. The bound is
subpower in H (only a logarithm). At frequency s, the L^2 spherical
integral against xi is O(1) per unit frequency, up to the cap losses.
Two exponents S = d/2 are being multiplied together to make d = 2S,
which is exactly the dimension. The problem is at the critical
exponent.
Proof sketch. The main idea is a Gaussian majorant + weighted Cauchy-
Schwarz.
Let lambda_B be the pushforward of the measure 1_{B <= |s| <= 2B} ds
d xi(w) under (s, w) -> s w. Then lambda_B is supported in the annulus
B <= |v| <= 2B and has total mass <= 2B. Its unit-ball concentration is
at most C L B^{-S}, from the cap bound on xi.
Let Phi be a rapidly decreasing, integrable kernel (a majorant for the
Fourier convolution of a smooth spatial cutoff). Then
|hat{alpha}(v)|^2 <= C int |hat{alpha}(u)|^2 Phi(v - u) du.
Integrating against lambda_B:
int |hat{alpha}|^2 d lambda_B <= C int |hat{alpha}(u)|^2 W_B(u) du,
with W_B = Phi * lambda_B. The unit-cube summation and decay of Phi give
W_B <= C L B^{-S} on all of R^d. And the tail estimate W_B(u) <= C_M
B (1 + |u|)^{-M} for |u| > 4B.
Now split the frequency integral into low and high parts. Low part:
Gaussian majorant for the ball indicator of |u| <= 4B. Uses the S-ball
bound to give
int_{|u| <= 4B} |hat{alpha}(u)|^2 du <= C K B^{d - S}.
High part: uses |hat{alpha}| <= 1 and the decay of W_B, which costs
O(1). Multiply the low part by W_B <= C L B^{-S}:
L K B^{d - 2S} <= L K B^{d - d} = L K.
So each dyadic band B contributes O(LK). Summing over B = 1, 2, 4,
..., H, that is O(LK log H). The final factor (1 + log H) replaces
log H explicitly for the B = 1 case. Done.
This is used many times. The two S's come from the two ball bounds:
one on alpha, one on xi. Their product is a constant per dyadic band.
That's the criticality of S = d/2.
5.6 Exceptional directions.
--------------------------------------------------------------------------------
We now prove that each test fails only on a small spherical set.
Lemma 5.2 (Exceptional directions). There is a dimensional constant
C_d (adjustable to be as large as needed) such that the following
holds for all sufficiently large N. For every spherical subprobability
xi satisfying the S-cap bound
xi(B(w, r)) <= R^sigma r^S for R^{-1} <= r <= 1,
every test [type, (g, p) or (p, t)] based at any occupied anchor cube
and any level zero satisfies
xi(S^{d-1} \ good_set_at_level_0) <= R^{-sigma}.
So the "good" set of directions has xi-measure at least 1 - R^{-sigma}.
Proof sketch for the parallel case. Divide directions into annuli by
distance to the anchor. In each annulus of scale 2^{-h}, the number of
depth-p centers at that distance is controlled by the profile via
I(h, p). A center at distance 2^{-h} from c_P is counted only for
directions in two caps of radius O(2^{-(p-h)} R^{500 sigma}). Cap
bound gives O(R^{(1+500S) sigma} 2^{-S(p-h)}).
Number of such centers <= C R^{I(h,p) + C_d' sigma}. Total expected
occupancy <= C (N+1) R^{I(h,p) - S(p-h)*N^{-1}... } -- the key identity
I(h, p) - S(p - h) = A(p) - A(h) <= k(g, p)
gives the total expected occupancy at most C (N+1) R^{k(g,p) + C_d''
sigma}. For fixed sigma, Markov at threshold R^{k(g,p) + C_d sigma}
gives the probability bound.
Proof sketch for the projection case. Rescale A to unit size. The
rescaled tolerance is eps = 2H^{-1} R^{500 sigma}. The function
F_eps(u) = (sin(u / (2 eps)) / (u / (2 eps)))^2
is a positive-definite majorant for 1_{|u| <= eps}, with Fourier
transform supported in [-eps^{-1}, eps^{-1}] = [-H/2, H/2] (times
2H^{-1}... careful with the constant). Bounded by C eps.
Then
int (alpha x alpha) { |(z - z').w| <= eps } d xi(w)
<= C eps int_{-H}^{H} int |hat{alpha}(s w)|^2 d xi(w) ds
<= C H^{-1} H^0 (log H) R^{delta_A + sigma} ...
The half-dimensional Fourier estimate (Lemma 5.1) with L = R^sigma and
K = R^{delta_A + C sigma} gives
int (...) d xi(w) <= C H^{-1} (log H) R^{A(p) - min_A + (501+C) sigma}.
Multiply by 1/H = 2^{-(t-p)} and absorb losses. Markov at the threshold
2^{-(t-p)} R^{A_p - min_A + C_d sigma} gives the bound. Done.
Both parts conclude the same way: at the level-zero test, the
"failure" set has xi-measure at most R^{-sigma}. This is our "cheap
exception" fact.
5.7 Smooth masks.
--------------------------------------------------------------------------------
We want a smooth envelope around the good-direction set, because we'll
later integrate against test indicators with derivatives. Fix a level
i < J_0. Define the "nominal angular scale"
D_{l'} = max(1, 2^{l'} R^{-50 sigma}),
where l' is the descriptor's nominal length (p-g for parallel, t-p for
projection).
Lemma 5.3 (Masks with a buffer). For all sufficiently large N, every
level i < J_0 admits a mask w_{i, tau}(x, w) in [0, 1], measurable in
x and smooth in w, such that on every occupied anchor cube:
- w = 1 on the level-i good set G_{i, tau}.
- supp_w w is contained in the level-(i+1) good set G_{i+1, tau}.
- Angular derivative bounds: || T_1 ... T_j w_{i, tau} ||_inf
<= C_j D_{l'}^j for every fixed collection of smooth tangent
fields T_j spanning the sphere's tangent space.
- The masks are antipodally symmetric (w(w) = w(-w)) and are
constant as functions of x on each anchor cube.
The mask construction: take a spherical convolution of the level-i
good-set indicator against a smooth rotationally invariant kernel of
support radius about c_0 / D_{l'}, with c_0 small. The buffer between
levels is D_{l'}^{-1}, and the smoothness index is preserved.
The point of the R^{500 sigma} vs R^{50 sigma} gap (which appears
between the width W_i and the mask smoothing scale) is that the
level-(i+1) good set strictly contains the D_{l'}^{-1}-neighborhood
of the level-i good set. That is what makes the buffer work. This
"500 vs 50" gap is small but positive, so it can be absorbed into the
exponents; but it must be respected throughout the recursion.
5.8 Products of masks.
--------------------------------------------------------------------------------
For any fixed-size list of descriptors tau_1, ..., tau_k (k = k(N)
bounded by N^{C_*} with C_* fixed), the product of the corresponding
masks is bounded in angular derivatives by (R^sigma D_max)^j where D_max
is max over nominal lengths in the list. The support of the product
still lies in the intersection of all the level-(i+1) supports of the
individual masks, and that is not lost by differentiation.
The "R^sigma" factor absorbs the polynomial blow-up from the product
rule. Since k <= N^{C_*} for a fixed C_*, expansion by the product
rule is bounded by a polynomial, and R^sigma dominates it for
sufficiently large N.
5.9 The fixed list M(a, t) for every depth interval.
--------------------------------------------------------------------------------
We need a family of descriptor lists that grows under interval
containment and is finite-size. Fix
q_0 = ceil(kappa N),
J_0 = 100(1/kappa + 1/sigma) + 100.
Partition {0, ..., N} into consecutive blocks of size q_0, with the
last possibly shorter. In each block, pick ONE depth minimizing
A. Call that the block's representative.
For an interval [a, t] with t - a > 100 q_0, pick
p = argmin_{n in a_* <= n <= t_*} A(n)
where [a_*, t_*] is the union of blocks strictly inside [a + q_0,
t - q_0]. Set l = min(p - a, t - p).
Call it a "prefix choice" if 2p <= a + t, "suffix choice" otherwise.
The list M(a, t): for every nonterminal subinterval [a', t'] subset
[a, t], include the parallel descriptors [g, p] for p - l <= g < p
where p, l is that interval's own policy choice. If the choice is a
suffix choice, also include the projection descriptor [p, t'].
De-duplicate. The resulting list is at most polynomially large in N
(for fixed sigma, kappa) and is contained in the corresponding lists
of larger intervals containing [a, t].
The "origin" of a descriptor in M(a, t) is one specific subinterval
[a', t'] that generates it. Every descriptor in the list comes from at
least one origin. This bookkeeping matters in Section 9's potential
argument.
5.10 Descriptor geometry.
--------------------------------------------------------------------------------
Lemma 5.4. For any nonterminal [a, t], the inner interval [a_*, t_*]
on which the policy chooses p satisfies
a + q_0 <= a_* <= a + 2 q_0,
t - 2 q_0 <= t_* <= t - q_0,
A(n) >= A(p) for all n in [a_*, t_*].
The lists M(a, t) are nested under interval containment. There are at
most 1 + (N+1)/q_0 possible anchors (one per block). For every
descriptor tau in M(a, t) with anchor p and nominal length l' = p - g,
there is an origin [a', t'] containing [p - l', p + l']. The same
origin supplies every parallel descriptor [g, p] for p - l' <= g < p.
The proof is direct bookkeeping. The inner interval is a strip of
width about 2q_0 on each side of [a, t]. Because the choice p minimizes
A over that strip, A is at least A(p) throughout the strip. Nesting of
the schedule follows from the fact that each block's choice is made
once and reused in every larger interval.
The "origin contains [p - l', p + l']" is exactly the fact that
prefix choices contribute l' = p - g <= p - a which is bounded by
half the origin's length; suffix choices are similar. So every
descriptor is "supported" by an interval at least twice its length,
positioned symmetrically around its anchor.