The non-covered set in Dvoretzky covering is a set of multiplicity

TL;DR

This work analyzes the non-covered set $E$ in Dvoretzky random covering as a multiplicity/ Rajchman problem. By constructing the random Dvoretzky measure $\\mu_D$ as the weak limit of densities $\\mu_n$ and examining the $d$-fold convolution $\\ast^d\\mu_D$, the authors establish conditions under which $\\ast^d\\mu_D$ becomes absolutely continuous, implying that $\\mu_D$ is Rajchman and $E$ is an $M_0$-set. The core method reduces to proving $L^1$ (and $L^2$) convergence of the $d$-fold density via a random-density convergence criterion, employing a decomposition around near-diagonal configurations with the kernel $K(t)=\\exp\left(\\sum_{k=1}^\infty(\\ell_k-|t|)_+\right)$. A key technical step is bounding the energy integral $L_n^{(\\delta)}$ and controlling the limit behavior of products of truncated indicators $P_k$, ultimately showing that the limiting density has Fourier coefficients vanishing at infinity (Rajchman), hence the non-covered set is $M_0$. The results apply to the canonical case $\\ell_n=\\alpha/n$ and connect to broader multiplicative chaos perspectives, including Gaussian multiplicative chaos.

Abstract

We prove that the non-covered set in Dvortezky random covering is a set of multiplicity, by showing that the natural multiplicative chaotic measure is a Rajchman measure.

Theorems & Definitions (17)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• Lemma 2.5
• proof