Continued fraction expansions in connection with the metric Mahler measure

TL;DR

This paper explores the relationship between metric Mahler measures and continued fractions, providing new methods to compute these measures in specific cases and advancing understanding of their properties.

Contribution

It establishes a novel connection between metric Mahler measures and continued fractions, enabling calculations in previously intractable cases.

Findings

Connected metric Mahler measures with continued fractions in special cases

Calculated new examples of metric Mahler measures

Provided insights into the structure of generalized metric Mahler measures

Abstract

The metric Mahler measure was first studied by Dubickas and Smyth in 2001 as a means of phrasing Lehmer's conjecture in topological language. More recent work of the author examined a parametrized family of generalized metric Mahler measures that gives rise to a series of new, and apparently difficult, problems. We establish a connection between these metric Mahler measures and the theory of continued fractions in a certain class of special cases. Our results enable us to calculate metric Mahler measures in several new examples.