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 $m_t(\alpha)$.
"Metric heights on an Abelian group." Rocky Mountain J. Math. 44 (6) 2075 - 2091, 2014. https://doi.org/10.1216/RMJ-2014-44-6-2075