Hausdorff dimension of the Cartesian product of exact approximation set in $\beta$-expansions

TL;DR

This paper investigates the Hausdorff dimension of Cartesian products of sets defined by exact approximation properties in different $eta$-expansions, extending metrical number theory to multidimensional settings.

Contribution

It provides a formula for the Hausdorff dimension of these product sets, generalizing previous results to multiple $eta$-expansion contexts.

Findings

Derived the Hausdorff dimension formula for product sets in $eta$-expansions.

Extended metrical approximation theory to multidimensional $eta$-expansion sets.

Established dimension results for sets approximable to specified orders.

Abstract

In this paper, we study the metrical theory of Cartesian products of exact approximation sets in $\beta$-expansions. More precisely, for an integer $d \ge 2$ and real numbers $\beta_i > 1$ $(1 \le i \le d)$, we consider the set of points $x_i \in [0,1)$ is approximable by its convergents in the $\beta_i$-expansion to order $\psi_i$, but not to any better order. For any non-increasing functions $\psi_i$, we determine the Hausdorff dimension of the Cartesian product of these sets.