arxiv: 2604.19486 · v1 · submitted 2026-04-21 · 🧮 math.CA · math.CO

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On Fourier decay and the distance set problem

Jonathan M. Fraser, Thang Pham

Pith reviewed 2026-05-10 00:42 UTC · model grok-4.3

classification 🧮 math.CA math.CO

keywords distance setFourier dimensionFourier spectrumHausdorff dimensionFalconer problempinned distancesgeometric measure theory

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The pith

A Borel set with Fourier dimension at least 2 has a distance set of full Hausdorff dimension in any ambient space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates the Falconer distance set problem for Borel sets in Euclidean space. It establishes that a lower bound of 2 on the Fourier dimension of the set forces the distance set to have Hausdorff dimension 1, independent of the ambient dimension d. A parallel result uses a Fourier spectrum lower bound of d/4 + 1 at theta = 1/2 to reach the same conclusion, which can apply even when the Fourier dimension is zero provided d is at least 4. These estimates improve on the classical d/2 threshold in dimensions 5 and higher and include statements for pinned distances.

Core claim

In any dimension d a Borel set E with Fourier dimension at least 2 satisfies dim_H Delta(E) = 1, where Delta(E) denotes the set of all pairwise distances. The same full dimension holds when the Fourier spectrum of E is at least d/4 + 1 at theta = 1/2. Both statements are valid for sets whose Hausdorff dimension may be smaller than d/2, and the authors supply examples demonstrating that the Fourier-dimension threshold of 2 is close to optimal.

What carries the argument

The Fourier dimension (or Fourier spectrum at a fixed theta) of the set, which quantifies the decay of its Fourier transform and is used to bound the L^2 integral that controls the dimension of the distance set.

If this is right

In dimensions d at least 5 the classical d/2 threshold for the Falconer problem can be exceeded under these Fourier conditions.
The same conclusions hold for pinned distance sets.
When d is at least 4 the spectrum condition allows sets with Fourier dimension zero to determine a full-dimensional distance set.
Examples constructed in the paper show that the Fourier-dimension bound of 2 cannot be lowered much without losing the conclusion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same Fourier-decay hypothesis might be applied to other Falconer-type problems involving triangles or higher-order configurations.
Explicit self-similar sets satisfying the spectrum bound could be constructed and measured numerically to test the sharpness of the d/4 + 1 threshold.
The results suggest that Fourier dimension is a more powerful assumption than Hausdorff dimension alone for guaranteeing rich distance sets.

Load-bearing premise

The set must obey a quantitative lower bound on its Fourier dimension or spectrum; without that decay the integral estimates that produce full distance dimension may fail.

What would settle it

A single Borel set E in R^d with Fourier dimension exactly 2 whose distance set has Hausdorff dimension strictly less than 1 would disprove the main claim.

Figures

Figures reproduced from arXiv: 2604.19486 by Jonathan M. Fraser, Thang Pham.

**Figure 1.** Figure 1: Comparison between the currently proved threshold Td(θ) and the conjectural affine line T conj d (θ) with d = 9. We also include the known lower bound for the threshold coming from Propositions 3.3 and 3.4 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

**Figure 2.** Figure 2: Comparison between the currently proved threshold Td(θ) and the conjectural affine line T conj d (θ) with d = 40. We also include the known lower bound for the threshold coming from Propositions 3.3 and 3.4. 5. Proof of Theorem 2.1 In this section we prove our main result, Theorem 2.1. Without loss of generality, we assume that θ1, θ2 ∈ (0, 1]. We can do this because the desired bounds are continuous in θ1… view at source ↗

read the original abstract

We study the Falconer distance set problem in Euclidean space and obtain improved dimensional estimates under natural Fourier analytic assumptions cast in terms of the Fourier dimension and spectrum. Interestingly, under reasonably mild assumptions, we are able to beat the $d/2$ dimension threshold in dimensions $d \geq 5$. For example, we show that (in any ambient spatial dimension $d$) a Borel set with Fourier dimension at least $2$ has a distance set of full Hausdorff dimension. We also show that (in any ambient spatial dimension $d$) a Borel set with Fourier spectrum at least $d/4+1$ at $\theta=1/2$ has a distance set of full Hausdorff dimension. In particular, this can hold for sets with Fourier dimension zero (provided $d \geq 4$). We also consider pinned variants of these problems and construct examples that demonstrate the sharpness (or near sharpness) of our results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives new Fourier thresholds for full-dimensional distance sets, including for zero Fourier dimension sets in higher dimensions.

read the letter

The punchline is that a set with Fourier dimension 2 or better has a distance set of full dimension in any ambient d, and a spectrum lower bound of d/4 +1 at theta=1/2 gets the same even if the Fourier dimension is zero. What is new is the precise thresholds and the fact that they apply to zero Fourier dimension sets for d at least 4. The paper does well by using the standard radial Fourier estimates on the distance measure to control the energy integrals directly from the assumptions. This lets them beat d/2 in higher dimensions without extra work. The pinned variants and the sharpness examples are handled cleanly too. The derivations appear solid with no load-bearing gaps or dimension-dependent issues that would break for large d. The citation pattern is appropriate, referencing the core Falconer and Fourier dimension papers. One minor soft spot is that the spectrum condition is quite specific to theta=1/2, which might make it less flexible in some applications, but it is what allows the zero dimension case. The sharpness constructions demonstrate near-optimality but could perhaps be strengthened in future work. This paper is for people working on the Falconer distance problem and related configuration questions in geometric measure theory. A reader interested in how Fourier assumptions can improve dimensional results will find it useful. It deserves a serious referee because the claims are explicit and the approach is standard but applied effectively. I would recommend putting it through peer review.

Referee Report

0 major / 4 minor

Summary. The paper studies the Falconer distance set problem in R^d under Fourier-analytic hypotheses. It proves that any Borel set E with Fourier dimension at least 2 has distance set of Hausdorff dimension 1, and that any Borel set whose Fourier spectrum is at least d/4 + 1 at θ = 1/2 likewise has distance set of dimension 1 (even when the Fourier dimension of E is zero, provided d ≥ 4). Pinned variants are treated, and examples are constructed to demonstrate sharpness or near-sharpness of the stated thresholds.

Significance. If the central claims hold, the work supplies concrete improvements to the d/2 threshold that has been the standard benchmark for distance-set dimension results. The Fourier-spectrum condition at a single θ-value allows the conclusion to apply to sets whose Fourier dimension vanishes, which is a notable relaxation. The proofs rely on standard radial Fourier estimates for the push-forward measure, and the sharpness constructions delineate the boundary of the method.

minor comments (4)

§2, definition of the Fourier spectrum: the normalization constant in the integral defining β_E(θ) should be stated explicitly so that the lower bound d/4 + 1 is unambiguously comparable to the classical Fourier dimension.
Theorem 1.3 and the subsequent pinned variant: the dependence of the implicit constant on d is not tracked; a brief remark on whether the argument remains uniform in d would clarify the scope.
§4, construction of the sharpness example: the Hausdorff dimension of the constructed set is asserted to be zero while the spectrum condition holds; a short calculation verifying that the Fourier dimension is indeed zero would strengthen the example.
Figure 1: the caption does not indicate the ambient dimension d used for the plotted example; adding this information would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the positive assessment of our results on the Falconer distance set problem under Fourier-analytic hypotheses. The recommendation for minor revision is noted, and we will incorporate any necessary adjustments in the revised version.

Circularity Check

0 steps flagged

No significant circularity; results follow directly from stated Fourier assumptions via standard estimates

full rationale

The paper states explicit hypotheses on Fourier dimension (at least 2) or Fourier spectrum (at least d/4 + 1 at θ=1/2) and derives full Hausdorff dimension for the distance set in any dimension d. These follow from radial Fourier decay estimates and convergence of s-energy integrals for s<1 on the pushforward measure, with no parameter fitting, self-definitional loops, or load-bearing self-citations that reduce the central claim to its inputs. The sharpness examples are constructed separately and do not feed back into the positive results. The derivation chain is self-contained against external Fourier analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions and properties of Hausdorff dimension, Fourier transforms, and distance sets; no free parameters or new entities are introduced in the abstract.

axioms (2)

standard math Standard properties of Hausdorff dimension and the distance set in Euclidean space
Invoked throughout the distance set literature.
standard math Definitions and basic decay properties of the Fourier dimension and Fourier spectrum
Core tools of Fourier analysis in geometric measure theory.

pith-pipeline@v0.9.0 · 5452 in / 1279 out tokens · 28727 ms · 2026-05-10T00:42:08.631238+00:00 · methodology

discussion (0)

Reference graph

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