Zero-full law for well approximable sets in missing digit sets

TL;DR

This paper investigates the approximation properties of numbers in missing digit sets within certain bases, establishing a zero-full law for Hausdorff measure and correcting previous errors in related theorems.

Contribution

It introduces a zero-full law for Hausdorff measure in missing digit sets, correcting prior work and extending results to new cases based on multiplicative dependence.

Findings

Established a zero-full law for Hausdorff measure in missing digit sets.

Corrected an error in previous theorems by Levesley, Salp, and Velani.

Provided bounds for Hausdorff dimension when bases are multiplicatively independent.

Abstract

Let $b \geq 3$ be an integer and $C(b,D)$ be the set of real numbers in $[0,1]$ whose base $b$ expansion only consists of digits in a set $D \subseteq \{0,...,b-1\}$. We study how close can numbers in $C(b,D)$ be approximated by rational numbers with denominators being powers of some integer $t$ and obtain a zero-full law for its Hausdorff measure in several circumstances. When $b$ and $t$ are multiplicatively dependent, our results correct an error of Levesley, Salp and Velani (Math. Ann., 338:97-118, 2007) and generalize their theorem. When $b$ and $t$ are multiplicatively independent but have the same prime divisors, we obtain a partial result on the Hausdorff measure and bounds for the Hausdorff dimension, which are close to the multiplicatively dependent case. Based on these results, several conjectures are proposed.