A winning approach to the intersections of twisted non-recurrent sets with fractals

Abstract

In this paper, we prove that if $S\subseteq\mathbb{R}^d$ is hyperplane absolute winning on a closed hyperplane diffuse set $L\subseteq\mathbb{R}^d$, then $\mathrm{dim}_H S\cap K=\mathrm{dim}_H K$ for any irreducible self-conformal set $K\subseteq L$ without assuming any separation condition on $K$. The result is then applied to obtain the Hausdorff dimension of intersections between irreducible self-conformal sets and twisted non-recurrent sets $\mathrm{N}(T,\mathcal{G})$ defined as $$ \mathrm{N}(T,\mathcal{G}):=\left\{\mathbf{x}\in[0,1]^d:\liminf_{n\to\infty}\|T^n(\mathbf{x})-g_n(\mathbf{x})\|>0\right\}, $$ where $T:[0,1]^d\to[0,1]^d$ belongs to a broad class of product maps, $\mathcal{G}:=\{g_n\}_{n\in\mathbb{N}}$ is a sequence of self-maps on $[0,1]^d$ with uniform Lipschitz constant and $\|\cdot\|$ denotes the maximal norm in $\mathbb{R}^d$. When $T$ is the $\beta$-transformation on $[0,1]$, it provides a positive answer to a question raised informally by Broderick, Bugeaud, Fishman, Kleinbock and Weiss (Math. Res. Lett., 2010). For the case $T$ is a $d\times d$ diagonal matrix transformations, our results provide a partial answer asked in a paper of Li, Liao, Velani and Zorin (Adv. Math., 2023). A natural generalization to non-autonomous setting is also obtained.