Small Mahler Measures From Digraphs

Abstract

We attach Mahler measures to digraphs and find combinatorial realizations of nearly all of the known low-degree ( $\le 180$) small (< 1.3) one-variable Mahler measures. We find one new such measure not on either of the lists maintained by Mossinghoff and Sac-Épée. Considering limits of sequences of measures attached to families of digraphs, we get combinatorial explanations for 57 of the 61 known irreducible two-variable measures below 1.37.

KEYWORDS:

• Mahler measure
• digraphs

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 11R06
• Secondary 05C22

Acknowledgments

We are extremely grateful for the careful reading of the manuscript by the referees, and for their helpful suggestions.

Declaration of Interest

No potential conflict of interest was reported by the author(s).