Metric Heights on an Abelian Group
Charles L. Samuels
TL;DR
The paper develops a general framework for metric heights on a multiplicative Abelian group by abstracting the Mahler measure into a family of t-metric heights φ_t for a base height φ. It shows that the function t ↦ φ_t(α) is piecewise governed by L_t norms of finite representations and that only finitely many t are exceptional, with uniform-replacement sets providing a finite description on intervals. It proves a network of equivalences linking attainment of the infimum, finite representations, and the existence of finite replacement sets, and demonstrates these results in the Mahler-measure context, resolving an open problem about m_t. The findings offer a broad, versatile toolkit for studying Lehmer-type questions and height structures on general Abelian groups, with concrete implications for number-theoretic heights and their metric variants.
Abstract
Suppose $m(α)$ denotes the Mahler measure of the non-zero algebraic number $α$. For each positive real number $t$, the author studied a version $m_t(α)$ 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 $m_t(α)$.