Metric Heights on an Abelian Group

Abstract

Suppose $m(\alpha )$ denotes the Mahler measure of the non-zero algebraic number $\alpha$. For each positive real number $t$, the author studied a version $m_t(\alpha )$ of the Mahler measure that has the triangle inequality. The construction of $m_t$ is generic, and may be applied to a broader class of functions defined on any Abelian group $G$. We prove analogs of known results with an abstract function on $G$ in place of the Mahler measure. In the process, we resolve an earlier open problem stated by the author regarding $m_t(\alpha )$.

References in corpus (5)

• An effective lower bound for the height of algebraic numbers
• On the non-Archimedean metric Mahler measure
• The parametrized family of metric Mahler measures
• The $t$-metric Mahler measures of surds and rational numbers
• The infimum in the metric Mahler measure

Cited by in corpus (4)

• 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