Rational points near self-similar sets

TL;DR

This paper investigates the distribution of rational points near self-similar sets, proving inhomogeneous approximation results and confirming conjectures for specific Cantor measures using Fourier analysis and number theory techniques.

Contribution

It establishes new equidistribution results for rational points near certain self-similar measures and partially proves a conjecture related to the middle-$p$ Cantor set.

Findings

Rational points are equidistributed near some self-similar measures.

Inhomogeneous Khinchine-type convergence results are proved for these measures.

Partial proof of a conjecture for the middle-$p$ Cantor set with $p>10^7$.

Abstract

In this paper, we consider a problem of counting rational points near self-similar sets. Let $n\geq 1$ be an integer. We shall show that for some self-similar measures on $\mathbb{R}^n$, the set of rational points $\mathbb{Q}^n$ is 'equidistributed' in a sense that will be introduced in this paper. This implies that an inhomogeneous Khinchine convergence type result can be proved for those measures. In particular, for $n=1$ and large enough integers $p,$ the above holds for the middle-$p$th Cantor measure, i.e. the natural Hausdorff measure on the set of numbers whose base $p$ expansions do not have digit $[(p-1)/2].$ Furthermore, we partially proved a conjecture of Bugeaud and Durand for the middle-$p$th Cantor set and this also answers a question posed by Levesley, Salp and Velani. Our method includes a fine analysis of the Fourier coefficients of self-similar measures together with an Erd\H{o}s-Kahane type argument. We will also provide a numerical argument to show that $p>10^7$ is sufficient for the above conclusions. In fact, $p\geq 15$ is already enough for most of the above conclusions.