The infimum in the metric Mahler measure

TL;DR

This paper proves that the infimum in the metric Mahler measure is always attained and establishes a similar result for the ultrametric Mahler measure, confirming a conjecture by Dubickas and Smyth.

Contribution

It confirms that the infimum in the metric Mahler measure definition is always achieved and extends this result to the ultrametric Mahler measure, resolving a conjecture.

Findings

The infimum in the metric Mahler measure is always attained.

An analogous result is established for the ultrametric Mahler measure.

The conjecture by Dubickas and Smyth is verified.

Abstract

Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number $\alpha$ by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved as well as establish its analog for the ultrametric Mahler measure.