Bounded L\"uroth 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 Hausdorff dimension, and satisfies 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 infinite Markov partition, which leads to the notion of infinite distortion (in the sense of Markov partitions).