The Lind-Lehmer Constant for $\mathbb Z_2^r \times \mathbb Z_{4}^s$

TL;DR

This paper determines the minimal positive logarithmic Lind-Mahler measure for groups of the form Z_2^r times Z_4^s, extending known results to new classes of 2-groups and providing explicit formulas.

Contribution

It establishes explicit formulas for the minimal Lind-Mahler measure for groups Z_2^r × Z_4^s and Z_2 × Z_{2^n}, extending previous knowledge to broader 2-group classes.

Findings

Minimal measure for Z_2^r × Z_4^s is (1/|G|) log(|G|-1)

For Z_2 × Z_{2^n} with n≥3, the measure is (1/|G|) log 9

Previous results only covered Z_2^k and Z_{2^k} groups

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 $|G|\geq 4$ is $\frac{1}{|G|} \log (|G|-1).$ We also show that for $G=\mathbb Z_2 \times \mathbb Z_{2^n}$ with $n\geq 3$ 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 $\mathbb Z_{2^k}.$