The parametrized family of metric Mahler measures
Charles L. Samuels
TL;DR
The paper develops a parametrized family of metric Mahler measures $M_t$ for algebraic numbers and explores how $t\mapsto M_t(\alpha)$ behaves, connecting it to Lehmer's conjecture via topologies on the quotient $V$ of the algebraic numbers by roots of unity. It proves a sharp equivalence: Lehmer's conjecture holds if and only if there exists a $t\ge1$ such that $M_t$ and $M_\infty$ induce the same topology on $V$, and shows that $\mu_\alpha(t)=M_t(\alpha)$ is piecewise given by $L^t$-norms of a finite set of vectors, with finitely many $\alpha$-exceptional points in any finite range. A rational-input conjecture (InfimumRational) posits that for rational $\alpha$, $M_t(\alpha)^t$ can be expressed as a sum of Mahler measures of rational surds, enabling a finite-factorization approach (RationalFactorization) to compute $M_t(\alpha)$ and study graphs of $\mu_\alpha$. The framework links height theory, metric geometry on multiplicative groups, and Lehmer-type questions, and offers practical finite-search tools for rational inputs.
Abstract
Let $M(α)$ denote the (logarithmic) Mahler measure of the algebraic number $α$. 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 $t\mapsto M_t(α)^t$ is constructed piecewise from certain sums of exponential functions. We pose a conjecture that, if true, enables us to graph $t\mapsto M_t(α)$ for rational $α$.