A collection of metric Mahler measures

TL;DR

This paper generalizes metric Mahler measures by introducing a family of measures parametrized by positive real numbers, analyzing their properties, and identifying minimality and continuity features.

Contribution

It extends previous constructions of metric Mahler measures to a continuous family, providing explicit computations and minimality properties.

Findings

Computed $M_x(eta)$ for small $x$ values.

Identified a minimality property of a certain function $ar M$.

Proved the continuity of $x o M_x(eta)$.

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.