The Lind-Lehmer Constant for ${\mathbb{Z}}_2^r\times {\mathbb{Z}}_4^s$

Abstract

We show that the minimal positive logarithmic Lind-Mahler measure for a group of the form $G={\mathbb{Z}}_2^r\times {\mathbb{Z}}_4^s$ with $|G|\ge 4$ is $\frac{1}{|G|}\log (|G|-1).$ We also show that for $G={\mathbb{Z}}_2\times {\mathbb{Z}}_{2^n}$ with $n\ge 3$ this value is $\frac{1}{|G|}\log 9.$ Previously the minimal measure was only known for $2$-groups of the form ${\mathbb{Z}}_2^k$ or ${\mathbb{Z}}_{2^k}.$

11 pages