The asymptotic Mahler measure of Gaussian periods

TL;DR

This paper investigates the asymptotic behavior of Mahler measures of Gaussian periods, revealing connections to log Calabi-Yau varieties and proposing conjectures about minimal Mahler measures among algebraic integers with cyclic Galois groups.

Contribution

It introduces a new sequence of cyclotomic integers with small Mahler measures, studies their asymptotics, and connects these to geometric and dynamical systems, proposing conjectures on minimal measures.

Findings

Growth rate of Mahler measures linked to log Calabi-Yau varieties.

Computational evidence suggests minimal Mahler measures for certain algebraic integers.

Conjectures indicate double logarithmic growth of Mahler measures with Galois group order.

Abstract

We construct a sequence of cyclotomic integers (Gaussian periods) of particularly small Mahler measure/height. We study the asymptotics of their Mahler measure as a function of their conductor, to find that the growth rate is the (multivariate) Mahler measure of a family of log Calabi-Yau varieties of increasing dimension. In turn, we study the asymptotics of some of these Mahler measures as the dimension increases, as well as properties of the associated algebraic dynamical system. We describe computational experiments that suggest that these cyclotomic integers realise the smallest non-zero logarithmic Mahler measure in the set of algebraic integers with cyclic Galois group of a given odd order. Finally, we discuss some precise conjectures that imply double logarithmic growth for those Mahler measures as a function of that order. The proofs use ideas from the theory of quantitative equidistribution, reflexive polytopes and toric varieties, the theory of random walks, Bessel functions, class field theory, and Linnik's constant.