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 $t\ge 1$. 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 $D\in \mathbb{N}$.

References in corpus (2)

• On the non-Archimedean metric Mahler measure
• The parametrized family of metric Mahler measures

Cited by in corpus (5)

• Metric Heights on an Abelian Group
• Continued fraction expansions in connection with the metric Mahler measure
• Counting Exceptional Points for Rational Numbers Associated to the Fibonacci Sequence
• Optimal factorizations of rational numbers using factorization trees
• Metric Mahler measures over number fields