Rotational beta expansions and Schmidt games

TL;DR

This paper studies rotational beta expansions across different dimensions and identifies conditions under which certain sets are winning or losing in Schmidt's game, linking dynamical systems with game theory.

Contribution

It introduces conditions for cylinder sets in rotational beta expansions to be winning or losing in Schmidt's game across multiple dimensions.

Findings

Conditions for winning sets in Schmidt's game are established.

Results apply to expansions in real numbers, complex numbers, and quaternions.

Provides a link between dynamical systems and Schmidt's game theory.

Abstract

We consider rotational beta expansions in dimensions 1, 2 and 4 and view them as expansions on real numbers, complex numbers, and quaternions, respectively. We give sufficient conditions on the parameters $\alpha, \beta \in (0,1)$ so that particular cylinder sets arising from the expansions are winning or losing Schmidt $(\alpha,\beta)$-game.