The winning property of mixed badly approximable numbers

TL;DR

This paper proves that the set of mixed badly approximable numbers with parameters summing to one is winning in Schmidt's game, advancing understanding of Diophantine approximation in p-adic and real contexts.

Contribution

It establishes that the set of mixed badly approximable numbers is 1/2-winning, improving previous results on the mixed Schmidt conjecture.

Findings

The set ad_p(i,j) is 1/2-winning in Schmidt's game.

This result extends the understanding of Diophantine approximation properties.

It advances the proof of the mixed Schmidt conjecture.

Abstract

For any pair of real numbers $(i,j)$ with $0<i,j<1$ and $i+j=1$, we prove that the set of $p$-adic mixed $(i,j)$-badly approximable numbers $\bad_p(i, j)$ is 1/2-winning in the sense of Schmidt's game. This improves a recent result of Badziahin, Levesley, and Velani on mixed Schmidt conjecture.