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

Abstract

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