A collection of metric Mahler measures

Abstract

Let $M(\alpha)$ denote the Mahler measure of the algebraic number $\alpha$. In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative group of algebraic numbers. Later, Fili and the author used similar techniques to study a non-Archimedean version. We show how to generalize the above constructions in order to associate, to each point in $(0,\infty]$, a metric version $M_x$ of the Mahler measure, each having a triangle inequality of a different strength. We are able to compute $M_x(\alpha)$ for sufficiently small $x$, identifying, in the process, a function $\bar M$ with certain minimality properties. Further, we show that the map $x\mapsto M_x(\alpha)$ defines a continuous function on the positive real numbers.