Exceptional Units and Numbers of Small Mahler Measure

Joseph Silverman

Abstract

Let α be a unit of degree d in an algebraic number field, and assume that α is not a root of unity. We conduct a numerical investigation that suggests that if α has small Mahler measure, there are many values of n for which 1 − α^{n} is a unit and also many values of m for which Φ_{m}(α) is a unit, where Φ_{m} is the m-th cyclotomic polynomial. We prove that the number of such values of n and m is bounded above by O(d^{1+0.7/ log log d}), and we describe a construction of Boyd that gives a lower bound of Ω(d^{0.6/ log log d}).