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

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 $D \in \mathbb N$.