Shrinking targets for non-autonomous dynamical systems corresponding to Cantor series expansions

TL;DR

This paper derives a Bowen-type formula for the Hausdorff dimension of shrinking target sets in non-autonomous systems related to Cantor series expansions, with applications to Diophantine approximation.

Contribution

It introduces a general formula for Hausdorff dimension in non-autonomous systems linked to Cantor series, expanding understanding of shrinking target problems.

Findings

Derived a closed-form Bowen type formula for Hausdorff dimension

Applied the formula to various examples including Diophantine approximation

Extended the theory to a broad class of non-autonomous dynamical systems

Abstract

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the non-autonomous dynamical system on the interval $[0,1)$, viewed as $\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilizing our theorem. In particular, we provide a Diophantine approximation interpretation of our scheme.