The mixed Schmidt conjecture in the theory of Diophantine approximation

TL;DR

This paper proves that a certain set related to the mixed Schmidt conjecture in Diophantine approximation is winning and has full dimension, supporting the conjecture's validity within the framework of the mixed Littlewood conjecture.

Contribution

It establishes the winning property and full dimension of a set connected to the mixed Schmidt conjecture, advancing understanding in Diophantine approximation.

Findings

The set is one quarter winning in Schmidt games.

The intersection of countably many such sets has full dimension.

Supports the mixed Littlewood conjecture framework.

Abstract

Let $\mathcal{D}=(d_n)_{n=1}^\infty$ be a bounded sequence of integers with $d_n\ge 2$ and let $(i, j)$ be a pair of strictly positive numbers with $i+j=1$. We prove that the set of $x \in \RR$ for which there exists some constant $c(x) > 0$ such that \[ \max\{|q|_\DDD^{1/i}, \|qx\|^{1/j}\} > c(x)/ q \qquad \forall q \in \NN \] is one quarter winning (in the sense of Schmidt games). Thus the intersection of any countable number of such sets is of full dimension. In turn, this establishes the natural analogue of Schmidt's conjecture within the framework of the de Mathan-Teuli\'e conjecture -- also known as the `Mixed Littlewood Conjecture'.