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arXiv · 1408.4166

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

Charles L. Samuels·Jonas Jankauskas
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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 $D \in \mathbb N$.

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BibTeXRIS

Charles L. Samuels, Jonas Jankauskas. 2014-08-18. The $t$-metric Mahler measures of surds and rational numbers. https://doi.org/10.1007/s10474-011-0126-y

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