The Erd\H{o}s similarity conjecture and Rajchman measures

Abstract

Let $A\subseteq\mathbb{R}$ support a probability measure whose Fourier--Stieltjes transform tends to zero at infinity. We prove that, for every $\varepsilon\in(0,1)$, there is a closed, $1$-periodic, nowhere dense set $E\subseteq\mathbb{R}$ such that \[ m(E\cap I)\ge1-\varepsilon \] for every interval $I$ of length $1$, while $E$ contains no affine copy of $A$. Thus every set supporting a Rajchman measure satisfies the Erd\H{o}s similarity conjecture in a uniform large-set form. The proof combines equidistribution modulo one for large dilates of the measure with a multiscale family of low-density periodic blockers; no quantitative rate of Fourier decay is used. We also refine a classical theorem of Iva\v{s}ev-Musatov, showing that for every Hausdorff gauge $h$ there is an $h$-null compact Rajchman support $K$ satisfying \[ \overline{\dim}_{\mathrm B}^{\log} K=\dim_{\mathrm P}^{\log} K=1. \] The value $1$ is sharp for both dimensions.