Dimension drop for intersections of Cantor sets

Abstract

Let $E\subset \mathbb{R}$ be a self-similar set generated by a homogeneous iterated function system $\hat{I}¦$ with contraction ratio $\ddot{I}\in (0,1)$. Assume that $\hat{I}¦$ satisfies the open set condition and ${\dim }_{\mathrm{H}}E<1$. Let $f$ be a $C^1$-diffeomorphism on $\mathbb{R}$. We prove that if $\log |f^{\prime }(x)|/\log \ddot{I}\notin \mathbb{Q}$ for every $x\in E\cap f^{-1}(E)$, then the upper Minkowski dimension of $f(E)\cap E$ is strictly less than the Hausdorff dimension of $E$. We also establish a quantitative dimension drop result when $E$ is a missing-digit set and $f$ is an affine map with rational slope satisfying a certain arithmetic condition. Based on these results and a result of Shmerkin [Ann. of Math., 2019], we obtain characterizations of ${\hat{I}}^3$ in various contexts such that ${\overline{\dim }}_{\mathrm{M}}\left(({\hat{I}}^{3}E+\hat{I}\pm )\cap E\right)<{\dim }_{\mathrm{H}}E$ for every $\hat{I}\pm \in \mathbb{R}$.