Metric Mahler measures over number fields

For an algebraic number α, the metric Mahler measure ({m_1(alpha)}) was first studied by Dubickas and Smyth [4] and was later generalized to the t-metric Mahler measure ({m_t(alpha)}) by the author [16]. The definition of ({m_t(alpha)}) 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(alpha)}) is attained by rational points when ({alphain 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.