The $t$-metric Mahler measures of surds and rational numbers

TL;DR

This paper studies the $t$-metric Mahler measure, proving that for rational numbers the infimum is attained with rational points, while providing counterexamples for algebraic numbers, and deriving explicit formulas for certain algebraic measures.

Contribution

It proves that for rational numbers, the $t$-metric Mahler measure's infimum is achieved with rational points, confirming a conjecture, and introduces explicit formulas for quadratic cases.

Findings

Infimum in $M_t(eta)$ for rationals is attained with rational points.

Counterexamples exist for algebraic numbers where the infimum is not attained.

Explicit formula for $M_t(D^{1/k})$ for square-free $D$.

Abstract

A. Dubickas and C. Smyth introduced the metric Mahler measure $$ M_1(\alpha) = \inf\left\{\sum_{n=1}^N M(\alpha_n): N \in \mathbb N, \alpha_1 \cdots \alpha_N = \alpha\right\}, $$ where $M(\alpha)$ denotes the usual (logarithmic) Mahler measure of $\alpha \in \overline{\mathbb Q}$. This definition extends in a natural way to the $t$-metric Mahler measure by replacing the sum with the usual $L_t$ norm of the vector $(M(\alpha_1), \dots, M(\alpha_N))$ for any $t\geq 1$. For $\alpha \in \mathbb Q$, we prove that the infimum in $M_t(\alpha)$ may be attained using only rational points, establishing an earlier conjecture of the second author. We show that the natural analogue of this result fails for general $\alpha\in\overline{\mathbb Q}$ by giving an infinite family of quadratic counterexamples. As part of this construction, we provide an explicit formula to compute $M_t(D^{1/k})$ for a square-free $D \in \mathbb N$.