Exact Diophantine approximation$\colon$ the simultaneous case in $\mathbb{R}^{2}$

TL;DR

This paper advances the understanding of the Hausdorff dimension of sets related to exact Diophantine approximation in two-dimensional real space, addressing a previously unresolved gap in the field.

Contribution

It provides a precise determination of the Hausdorff dimension for the set of exact approximation orders in the simultaneous case in .

Findings

Resolved a gap in the Hausdorff dimension study for exact approximation sets

Established new results on the dimension in  for simultaneous approximation

Contributed to the theoretical understanding of Diophantine approximation

Abstract

We fill a gap in the study of the Hausdorff dimension of the set of exact approximation order considered by Fregoli [Proc. Amer. Math. Soc. 152 (2024), no. 8, 3177--3182].