Rational points near self-similar sets

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.