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

Abstract

A. Dubickas and C. Smyth introduced the metric Mahler measure $$ M_1(α) = \inf\left\{\sum_{n=1}^N M(α_n): N \in \mathbb N, α_1 \cdots α_N = α\right\}, $$ where $M(α)$ denotes the usual (logarithmic) Mahler measure of $α\in \overline{\mathbb Q}$. 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 $(M(α_1), \dots, M(α_N))$ for any $t\geq 1$. For $α\in \mathbb Q$, we prove that the infimum in $M_t(α)$ 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 $α\in\overline{\mathbb Q}$ 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 $D \in \mathbb N$.

Theorems & Definitions (16)

• Conjecture 1.1: Lehmer's conjecture
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof : Proof of Lemma \ref{['NewSequence']}
• proof : Proof of Theorem \ref{['RationalAttained']}
• Lemma 3.1