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Dimension of Diophantine approximation and some applications in harmonic analysis

Longhui Li, Bochen Liu

TL;DR

The paper builds a versatile, parameterized family of Diophantine-approximation sets in Euclidean space and develops a coherent framework linking Hausdorff and Fourier dimensions to harmonic-analytic phenomena. It provides a unified interpolation between classical limsup/liminf constructions, constructs measures that realize prescribed dimensions, and applies these to sharp projection bounds, ABC sum-product, and Fourier-restriction theory in both geometric and non-geometric regimes. Key contributions include explicit constructions achieving dim_H and dim_F pairs (s,t), sharp bounds for projections and sum-product in the plane and higher dimensions, and a complete picture of sharp Fourier-restriction phenomena tied to dimension theory. The results have broad implications for the understandings of how fractal geometry interacts with Fourier-analytic decay and restriction phenomena, offering explicit, verifiable examples and a robust methodological framework.

Abstract

In this paper we construct a new family of sets based on Diophantine approximation in the Euclidean space, and consider their applications in several problems in harmonic analysis. Our first application is on the Hausdorff dimension of our sets. We show a recent result of Ren and Wang on the ABC sum-product problem is sharp. Higher dimensional cases and the relation to orthogonal projections are also discussed. Some conjectures are proposed. In addition to Hausdorff dimension, we also consider Fourier dimension. For every $0\leq t\leq s\leq 1$, we are able to construct a subset of $\mathbb{R}$ that has Hausdorff dimension $s$ and Fourier dimension $t$, together with a measure $μ$ that captures both dimensions, i.e., $$μ(B(x,r))\lesssim_εr^{s-ε} \ \text{and} \ |\hatμ(ξ)|\lesssim_ε|ξ|^{-t/2 +ε}, \ \forall\,ε>0.$$ It is fundamental but the very first such result in the literature. Our last result is to provide a viewpoint of the sharpness of Fourier restriction over general measures from dimensions of sets and measures.

Dimension of Diophantine approximation and some applications in harmonic analysis

TL;DR

The paper builds a versatile, parameterized family of Diophantine-approximation sets in Euclidean space and develops a coherent framework linking Hausdorff and Fourier dimensions to harmonic-analytic phenomena. It provides a unified interpolation between classical limsup/liminf constructions, constructs measures that realize prescribed dimensions, and applies these to sharp projection bounds, ABC sum-product, and Fourier-restriction theory in both geometric and non-geometric regimes. Key contributions include explicit constructions achieving dim_H and dim_F pairs (s,t), sharp bounds for projections and sum-product in the plane and higher dimensions, and a complete picture of sharp Fourier-restriction phenomena tied to dimension theory. The results have broad implications for the understandings of how fractal geometry interacts with Fourier-analytic decay and restriction phenomena, offering explicit, verifiable examples and a robust methodological framework.

Abstract

In this paper we construct a new family of sets based on Diophantine approximation in the Euclidean space, and consider their applications in several problems in harmonic analysis. Our first application is on the Hausdorff dimension of our sets. We show a recent result of Ren and Wang on the ABC sum-product problem is sharp. Higher dimensional cases and the relation to orthogonal projections are also discussed. Some conjectures are proposed. In addition to Hausdorff dimension, we also consider Fourier dimension. For every , we are able to construct a subset of that has Hausdorff dimension and Fourier dimension , together with a measure that captures both dimensions, i.e., It is fundamental but the very first such result in the literature. Our last result is to provide a viewpoint of the sharpness of Fourier restriction over general measures from dimensions of sets and measures.
Paper Structure (25 sections, 11 theorems, 237 equations, 1 figure)

This paper contains 25 sections, 11 theorems, 237 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hausdorff dimension of Diophantine approximation
  3. Orthogonal projection and sum-product
  4. In the plane
  5. Higher dimensions
  6. Fourier dimension of Diophantine approximation
  7. Sharpness of Fourier restriction estimates
  8. The geometric case
  9. The non-geometric case
  10. Dimension of measures
  11. Hausdorff dimension of Diophantine approximation
  12. $\gamma=0$
  13. $\gamma>0$
  14. Orthogonal projection and sum-product
  15. In the plane
  16. ...and 10 more sections

Key Result

Theorem 1.1

Suppose $\gamma,\beta_1,\dots,\beta_d\geq 0$ and $0<\gamma+\beta_j<1$ for all $j$. Then there exists an increasing $\{q_i\}$ in $\mathbb{R}_+$ such that the set has Hausdorff dimension In particular it is sufficient to take $\{q_i\}$ as an increasing sequence in $(1,\infty)$ satisfying

Figures (1)

  • Figure 1: $\frac{\sum s_i+t}{3}$ and $f(s_1, s_2, s_3, t)$ as functions of $t$

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 10 more