Bounded L\"uroth expansions: applying Schmidt games where infinite distortion exists

TL;DR

This paper proves that the set of numbers with bounded Lüroth expansions is winning and strong winning, implying it is dense, has full Hausdorff dimension, and satisfies intersection properties, even with infinite distortion.

Contribution

It extends Schmidt game techniques to Lüroth expansions with infinite distortion, establishing their bounded expansion sets as winning and strong winning.

Findings

The set of bounded Lüroth expansion numbers is winning.

The set has full Hausdorff dimension.

The set is dense and satisfies countable intersection properties.

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).