On the Metric Mahler Measure

Abstract

We define a new height function on the group of non-zero algebraic numbers α, the height of α being the infimum over all products of Mahler measures of algebraic numbers whose product is α. We call this height the metric Mahler measure, since its logarithm defines a metric in the factor group of the non-zero algebraic numbers modulo the group of all roots of unity. This metric induces the discrete topology on this factor group if and only if Lehmer's conjecture is true. Sharp upper and lower bounds are obtained for the metric Mahler measure of an algebraic number in terms of its Mahler measure, degree and house. The metric Mahler measure is computed for a class of numbers that includes Salem and Pisot numbers, and for roots of rationals. We also show that the set of all ratios of the logarithms of the metric and classical Mahler measures of algebraic numbers having a fixed degree is everywhere dense in the interval given by these bounds.