Bounded Lüroth expansions: applying Schmidt games where infinite distortion exists

Abstract

We show that the set of numbers with bounded L uroth expansions (or bounded L uroth series) is winning and strong winning. From either winning property, it immediately follows that the set is dense, has full Hausdor dimension, and satis es a countable intersection property. Our result matches the well-known analogous result for bounded continued fraction expansions or, equivalently, badly approximable numbers. We note that L uroth expansions have a countably in nite Markov partition, which leads to the notion of in nite distortion (in the sense of Markov partitions).