The Hausdorff dimension of the intersection of $\psi$-well approximable numbers and self-similar sets

Abstract

Let $\psi :\mathbb{N}\to {\mathbb{R}}_{+}$ be a monotonically non-increasing function, and let ${\psi }_v:\mathbb{N}\to {\mathbb{R}}_{+}$ be defined by ${\psi }_v(q)=1/q^v$. In this article, we consider self-similar sets whose iterated function systems satisfy the open set condition. For functions $\psi$ that do not decrease too rapidly, we give a conjecturally sharp upper bound on the Hausdorff dimension of the intersection of $\psi$-well approximable numbers and such self-similar sets. When $\psi ={\psi }_v$ for some $v$ greater than 1 and sufficiently close to $1$, we give a lower bound for this Hausdorff dimension, which asymptotically matches the upper bound as $v\downarrow 1$. In particular, we show that the set of very well approximable numbers has full Hausdorff dimension within self-similar sets, thus confirming a conjecture of Levesley, Salp, and Velani.

22 pages