METRIC HEIGHTS ON AN ABELIAN GROUP

Suppose m(alpha) denotes the Mahler measure of the non-zero algebraic number a. 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 mt 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).