Weighted badly approximable vectors and games

TL;DR

This paper proves that certain sets of badly approximable vectors, defined by specific weight conditions, are winning sets in the context of Diophantine approximation, highlighting their robustness and significance.

Contribution

It establishes that the set of (r_1,...,r_d)-badly approximable vectors is winning when the first d-1 weights are equal and at least as large as the last, extending previous results.

Findings

The set of (r_1,...,r_d)-badly approximable vectors is winning under specified weight conditions.

Winning sets have full Hausdorff dimension in the relevant space.

The result applies to a class of weighted Diophantine approximation problems.

Abstract

We prove that the set of (r_1,r_2,..,r_{d})-badly approximable vectors is a winning set if r_1=r_2=...=r_{d-1}\geq r_{d}.