Metric Mahler measures over number fields

Abstract

For an algebraic number $α$, the metric Mahler measure $m_1(α)$ was first studied by Dubickas and Smyth in 2001 and was later generalized to the $t$-metric Mahler measure $m_t(α)$ by the author in 2010. The definition of $m_t(α)$ involves taking an infimum over a certain collection $N$-tuples of points in $\overline{\mathbb Q}$, and from previous work of Jankauskas and the author, the infimum in the definition of $m_t(α)$ is attained by rational points when $α\in \mathbb Q$. As a consequence of our main theorem in this article, we obtain an analog of this result when $\mathbb Q$ is replaced with any imaginary quadratic number field of class number equal to $1$. Further, we study examples of other number fields to which our methods may be applied, and we establish various partial results in those cases.