Metric Heights on an Abelian Group

TL;DR

This paper introduces a generalized version of the Mahler measure with triangle inequality on Abelian groups, extending known results and resolving an open problem about this new measure.

Contribution

It develops a generic construction of a measure-like function on Abelian groups that satisfies the triangle inequality, generalizing Mahler measure.

Findings

Established analogs of classical Mahler measure results for the new measure

Resolved an open problem regarding the properties of the constructed measure

Demonstrated the broad applicability of the construction to various functions on Abelian groups

Abstract

Suppose $m(\alpha)$ denotes the Mahler measure of the non-zero algebraic number $\alpha$. For each positive real number $t$, the author studied a version $m_t(\alpha)$ of the Mahler measure that has the triangle inequality. The construction of $m_t$ is generic, and may be applied to a broader class of functions defined on any Abelian group $G$. We prove analogs of known results with an abstract function on $G$ in place of the Mahler measure. In the process, we resolve an earlier open problem stated by the author regarding $m_t(\alpha)$.