On Fourier decay and the distance set problem

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 $θ=1/2$ 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.

• The paper establishes improved lower bounds for the Hausdorff dimension of distance sets by leveraging refined Fourier spectrum conditions.
• It demonstrates that under natural Fourier decay properties, even higher-dimensional sets achieve full distance set dimension, including pinned and biset variants.
• The study bridges L∞ and L² Fourier dimensions, providing sharp criteria that extend the classical Falconer distance conjecture under measurable Fourier constraints.

On Fourier Decay and the Distance Set Problem

Introduction and Problem Overview

The paper "On Fourier decay and the distance set problem" (2604.19486) advances the analysis of the Falconer distance set problem by employing refined Fourier analytic invariants, notably the Fourier spectrum, to obtain improved lower bounds on the Hausdorff dimension of distance sets. The Falconer conjecture posits that if a Borel set $E \subset \mathbb{R}^d$ has $\dim_{\mathrm{H}} E > d/2$, then $D(E)$---the set of Euclidean distances realized within $E$---should have full Hausdorff dimension (i.e., equals 1). Classical constructions show that the $d/2$ threshold cannot be improved solely under dimensional assumptions on $E$; notably, sets of small Fourier dimension fail to satisfy the conjectured bound. However, this paper demonstrates that many rigid counterexamples exhibit poor Fourier decay, motivating the use of additional Fourier-analytic hypotheses.

The authors develop dimensional results expressed in terms of the Fourier dimension and, more generally, the Fourier spectrum, a measurement of Fourier decay rates in various $L^p$-regimes. Key innovations include the beating of the $d/2$ threshold in higher dimensions under natural and checkable Fourier analytic conditions, the development of effective criteria in terms of the Fourier spectrum for full dimension and positive Lebesgue measure of the distance set, and the analysis of both standard and pinned variants of the problem.

Main Results

Beating the $d/2$ Threshold via Fourier Spectrum

A principal contribution is the derivation of improved lower bounds for the Hausdorff dimension of the distance set (and pinned versions) under Fourier spectrum constraints. Notably, the paper establishes that for $d \geq 5$, if $\dim_{\mathrm{H}} E > d/2$0 is a Borel set with $\dim_{\mathrm{H}} E > d/2$1 (i.e., the supremum over supported measures), then $\dim_{\mathrm{H}} E > d/2$2 attains full Hausdorff dimension. More generally, they show that if a Borel set has Fourier spectrum at least $\dim_{\mathrm{H}} E > d/2$3 at $\dim_{\mathrm{H}} E > d/2$4, i.e., $\dim_{\mathrm{H}} E > d/2$5, then $\dim_{\mathrm{H}} E > d/2$6 also achieves full dimension, regardless of the set's pure Fourier dimension (which may be zero for $\dim_{\mathrm{H}} E > d/2$7). This structurally expands the class of sets for which Falconer-type conclusions can be drawn, far beyond Salem sets.

Concretely, the authors prove that:

• If a compactly supported probability measure $\dim_{\mathrm{H}} E > d/2$8 on $\dim_{\mathrm{H}} E > d/2$9 satisfies $D(E)$0, then for $D(E)$1-almost all $D(E)$2, $D(E)$3 has full Hausdorff dimension.
• For any $D(E)$4, there exist explicit thresholds $D(E)$5 such that if $D(E)$6, then $D(E)$7. The explicit form of $D(E)$8 (see Corollary 3 in the paper) interpolates between known critical values and reduces to familiar thresholds at $D(E)$9 and $E$0.

Pinned Distance and Biset Variants

The paper extends these results to pinned distance sets $E$1 and biset distances $E$2. Strong lower bounds on their dimension and Lebesgue measure are established for almost every "pin" $E$3 and uniformly over compact sets $E$4 whose supporting measures exhibit prescribed Fourier spectrum properties. In particular, the authors show for a large class of measures (those with sufficiently large spectrum at $E$5) that the conclusions are near-optimal, and construct explicit examples to establish sharpness.

Connections to Potential Theory and Interpolation

The results hinge crucially on a fine interpolation between the $E$6-controlled Fourier dimension and the $E$7-controlled Hausdorff (energy) dimension, measured by the Fourier spectrum, as introduced by Fraser. The analysis smoothly interpolates the transition of threshold values along the $E$8 parameter and yields quantitative bounds that, in several regimes, outperform previous best-known results.

Examples and Sharpness

The paper systematically constructs counterexamples and extremizers that witness the near-sharpness or true sharpness of their lower bounds. In particular:

• Product sets with controlled Fourier spectrum but minimal distance sets ($E$9 is a singleton) demonstrate the necessity of the presented thresholds.
• For $d/2$0, the lower bound $d/2$1 is provably sharp: there exist sets of spectrum exactly $d/2$2 and dimension arbitrarily close to $d/2$3 for which $d/2$4 fails to have dimension 1.
• These sharpness results confirm that the critical thresholds found using the Fourier spectrum cannot, in general, be further reduced.

Implications and Theoretical Significance

Falconer Distance Problem Under Fourier Constraints

The implications for the geometric measure theory community are clear: under mild, natural, and verifiable Fourier decay conditions (quantified by the Fourier spectrum), the Falconer conjecture can be resolved with thresholds strictly below $d/2$5 in high dimensions, expanding the roster of `dimensionally large' sets for which the conjecture is valid.

Fourier Spectrum as a Canonical Interpolant

This work reinforces the utility of the Fourier spectrum as the correct analytic calibrator between completely rigid sets (which typically have poor Fourier decay) and Salem-type sets (with optimal decay), capturing subtle arithmetical and geometric information necessary in nonlinear projection and distance set problems.

Cross-Connections and Future Directions

The mapping of the optimal dimension threshold as a function of the Fourier spectrum exponent $d/2$6 hints at deeper structural phenomena. The paper conjectures an affine threshold $d/2$7 as the optimal criterion for full dimension in all intermediate regimes, interpolating exactly between natural endpoints. This model is informed partly by finite field analogs and by heuristic projections, and its resolution may yield profound consequences both for the Falconer problem and related harmonic analysis questions (such as sharp restriction phenomena).

Practically, these techniques suggest new Fourier-analytic strategies for problems involving nonlinear projections, additive combinatorics, and fractal percolation, and point towards the utility of spectrum-based analysis in quantifying the impact of `arithmetic rigidity' on geometric and analytic properties.

Conclusion

"On Fourier decay and the distance set problem" introduces a framework that quantifies distance set dimension via the Fourier spectrum, achieving new and in some regimes optimal bounds that beat classical thresholds under natural analytic assumptions. The work extends the paradigm from rigid notion of dimension to more refined Fourier analytic invariants, and its techniques and results suggest significant avenues for both theoretical progress and applications in geometric measure theory and harmonic analysis. It also clearly elucidates the role of structure versus randomness in the Falconer distance problem, and provides a comprehensive map of currently achieved and conjectured thresholds as parameterized by the Fourier spectrum.