Intersecting well approximable and missing digit sets

Abstract

Let $b\ge 3$ be an integer and $C(b,D)$ be the set of real numbers in $[0,1]$ whose $b$-ary expansion consists of digits restricted to a given set $D\subseteq \{0,\dots ,b-1\}$. Given an integer $t\ge 2$ and a real, positive function $\ddot{I}ˆ$, let $W_t(\ddot{I}ˆ)$ denote the set of $x$ in $[0,1]$ for which $|x-p/t^n|<\ddot{I}ˆ(n)$ for infinitely many $(p,n)\in \mathbb{Z}\times \mathbb{N}$. We prove a general Hausdorff dimension result concerning the intersection of $W_t(\ddot{I}ˆ)$ with an arbitrary self similar set which implies that ${\dim }_{\mathrm{H}}(W_t(\ddot{I}ˆ)\cap C(b,D))\le {\dim }_{\mathrm{H}}{W}_t(\ddot{I}ˆ)\times {\dim }_{\mathrm{H}}C(b,D)$. When $b$ and $t$ have the same prime divisors, under certain restrictions on the digit set $D$, we give a sufficient condition for the Hausdorff measure of $W_t(\ddot{I}ˆ)\cap C(b,D)$ to be zero. This closes a gap in a result of Li, Li and Wu \cite{LLW2025} and shows that the dimension of the intersection can be strictly less than the product of the dimensions. The latter disproves the product conjecture of Li, Li and Wu.

31 pages