On the Hausdorff Dimension of weighted exactly Approximable Vectors

TL;DR

This paper proves that the Hausdorff dimension of weighted exactly approximable vectors matches that of weighted approximable vectors in real space, extending previous results in Diophantine approximation.

Contribution

It generalizes earlier work by showing the equality of Hausdorff dimensions for weighted exactly approximable and approximable vectors, broadening understanding in metric number theory.

Findings

Hausdorff dimension of weighted exactly approximable vectors equals that of weighted approximable vectors

Generalization of previous results by the first author and De Saxcé

Extends the theory of Diophantine approximation in weighted settings

Abstract

We show that the Hausdorff dimension of $\boldsymbol w$-weighted $\tau$-exactly approximable vectors in $\mathbb R^d$ coincides with the Hausdorff dimension of $\boldsymbol w$-weighted $\tau$-approximable vectors, generalizing a result of the first named author and De Saxc\'e.