The parametrized family of metric Mahler measures

Abstract

Let $M(\alpha )$ denote the (logarithmic) Mahler measure of the algebraic number $\alpha$. Dubickas and Smyth, and later Fili and the author, examined metric versions of $M$. The author generalized these constructions in order to associate, to each point in $t\in (0,\infty ]$, a metric version $M_t$ of the Mahler measure, each having a triangle inequality of a different strength. We further examine the functions $M_t$, using them to present an equivalent form of Lehmer's conjecture. We show that the function $t\mapsto M_t(\alpha {)}^t$ is constructed piecewise from certain sums of exponential functions. We pose a conjecture that, if true, enables us to graph $t\mapsto M_t(\alpha )$ for rational $\alpha$.

References in corpus (1)

• On the non-Archimedean metric Mahler measure

Cited by in corpus (6)

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