A collection of metric Mahler measures
Charles L. Samuels

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 denote the Mahler measure of the algebraic number . 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 , a metric version of the Mahler measure, each having a triangle inequality of a different strength. We are able to compute for sufficiently small , identifying, in the process, a function with certain minimality properties. Further, we show that the map defines a continuous function on the positive real numbers.
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Taxonomy
Topicsadvanced mathematical theories · Advanced Topology and Set Theory · Mathematical and Theoretical Analysis
