Intersecting well approximable and missing digit sets

TL;DR

This paper investigates the Hausdorff dimension of intersections between well-approximable sets and digit-restricted Cantor sets, establishing bounds, conditions for measure zero, and disproving a conjecture about dimension products.

Contribution

It provides a general dimension result for intersections of well-approximable sets with self-similar sets, and disproves the product conjecture of Li, Li, and Wu.

Findings

Dimension of intersection is at most the product of individual dimensions.

Under certain conditions, the Hausdorff measure of the intersection is zero.

The product conjecture of Li, Li, and Wu is disproved.

Abstract

Let $b\geq3$ 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 $D\subseteq\{0,\ldots,b-1\}$. Given an integer $t\geq2$ and a real, positive function $\psi$, let $W_{t}(\psi)$ denote the set of $x$ in $[0,1]$ for which $|x-p/t^{n}|<\psi(n)$ for infinitely many $(p,n)\in\mathbb{Z}\times\mathbb{N}$. We prove a general Hausdorff dimension result concerning the intersection of $W_{t}(\psi)$ with an arbitrary self similar set which implies that $\dim_{\rm H}(W_{t}(\psi)\cap C(b,D))\le\dim_{\rm H}W_{t}(\psi)\times \dim_{\rm H}C(b,D)$. 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 $W_{t}(\psi)\cap C(b,D)$ 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.