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

Michael J. Mossinghoff; Vincent Pigno; Christopher Pinner

doi:10.2140/moscow.2019.8.151

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

Number Theory 2019-05-29 v1

Authors: Michael J. Mossinghoff , Vincent Pigno , Christopher Pinner

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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 $\mathbb Z_{2^k}.$

Keywords

Lind-Lehmer constant logarithmic Lind-Mahler measure finite abelian groups 2-groups group invariants

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@article{arxiv.1805.05450,
  title  = {The Lind-Lehmer Constant for $\mathbb Z_2^r \times \mathbb Z_{4}^s$},
  author = {Michael J. Mossinghoff and Vincent Pigno and Christopher Pinner},
  journal= {arXiv preprint arXiv:1805.05450},
  year   = {2019}
}

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11 pages

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

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