Rotational beta expansions and Schmidt games

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 $\hat{I}\pm ,{\hat{I}}^2\in (0,1)$ so that particular cylinder sets arising from the expansions are winning or losing Schmidt $(\hat{I}\pm ,{\hat{I}}^2)$-game.

33 pages. Published in Acta Mathematica Hungarica. Version 2: Revised in accordance with the referee report. In particular, we have corrected the deficiencies in the statement and proof of Theorem 4.1. Version 3: Corrections emphasizing completeness of metric space used