Dimension drop for intersections of Cantor sets

Let $E\subset \mathbb{R}$ be a self-similar set generated by a homogeneous iterated function system $\Phi$ with contraction ratio $\rho\in (0,1)$.

Assume that $\Phi$ satisfies the open set condition and $\dim_{\rm H}E<1$.

Let $f$ be a $C^1$-diffeomorphism on $\mathbb{R}$.

We prove that if $\log|f'(x)|/\log\rho\not\in\mathbb{Q}$ 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 $\gamma$ in various contexts such that $\overline{\dim}_{\rm M}\left((\gamma E+\alpha)\cap E\right)<\dim_{\rm H}E$ for every $\alpha\in\mathbb{R}$.