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Salem properties of Dvoretzky random coverings

Yukun Chen, Xiangdi Fu, Zhaofeng Lin, Yanqi Qiu

TL;DR

The paper establishes that, for Dvoretzky random coverings with non-Shepp gaps and $D_\ell<1$, the uncovered set $K_\ell$ is almost surely a Salem set with $\dim_{\mathcal{F}} K_\ell=\dim_{\mathcal{H}} K_\ell=1-D_\ell$, conditional on a nondegenerate multiplicative chaos measure $\mu_{RC}$. It develops a vector-valued martingale framework to connect Fourier decay to fractal dimensions and introduces a translation-cancellation trick along with $L^1$-modulus and mean-oscillation controls to manage high-frequency Fourier terms in the absence of weak spatial independence. The analysis hinges on careful moment estimates for the martingale increments $D_k$, a dyadic frequency decomposition, and a zero-one law ensuring unconditional results. Overall, the work extends Salem-type analyses to a setting where classical independence is unavailable, by combining probabilistic martingale methods with harmonic-analysis techniques for random fractals on the circle.

Abstract

We establish the Salem properties for the uncovered sets in the celebrated Dvoretzky random coverings of the unit circle.

Salem properties of Dvoretzky random coverings

TL;DR

The paper establishes that, for Dvoretzky random coverings with non-Shepp gaps and , the uncovered set is almost surely a Salem set with , conditional on a nondegenerate multiplicative chaos measure . It develops a vector-valued martingale framework to connect Fourier decay to fractal dimensions and introduces a translation-cancellation trick along with -modulus and mean-oscillation controls to manage high-frequency Fourier terms in the absence of weak spatial independence. The analysis hinges on careful moment estimates for the martingale increments , a dyadic frequency decomposition, and a zero-one law ensuring unconditional results. Overall, the work extends Salem-type analyses to a setting where classical independence is unavailable, by combining probabilistic martingale methods with harmonic-analysis techniques for random fractals on the circle.

Abstract

We establish the Salem properties for the uncovered sets in the celebrated Dvoretzky random coverings of the unit circle.

Paper Structure

This paper contains 18 sections, 9 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Dvoretzky random coverings
  3. Main result
  4. Proof sketch and new ingredients in Dvoretzky random coverings
  5. A quick review of the vector-valued martingale method.
  6. New ingredients
  7. Further discussions and remarks
  8. The multiplicative chaos measures for Dvoretzky random coverings
  9. Proof of Theorem \ref{['theorem-main']} conditioned on $\{\mu_\textnormal{RC}\ne 0\}$
  10. Pointwise upper estimate
  11. Dyadic frequency decomposition
  12. A variant of the translation-cancellation trick for high-frequency terms
  13. Moment estimates for the $L^1$-norm and $L^1$-modulus of continuity of $D_k$.
  14. Conclude Proposition \ref{['theorem-submain']}
  15. Proof of Theorem \ref{['theorem-main']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Assuming eq-non-Shepp, the uncovered set $K_\ell$ is almost surely Salem with Fourier dimension

Theorems & Definitions (22)

  • Remark 1.1
  • Theorem 1.1
  • Proposition 1.2
  • proof : Proof of Proposition \ref{['prop-vector-valued-martingale']}
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • ...and 12 more