Exceptional Units and Numbers of Small Mahler Measure

Abstract

Let α be a unit of degree d in an algebraic number field, and assume that a is not a root of unity. We conduct a numerical investigation that suggeststhat 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 nand m is bounded above by O(d ^{l+0.7/log log d} ), and we describe a construction of Boyd that givesa lower bound of Ω(d ^{0.6/log log d} ).

Keywords:

• units
• Mahler measure
• unit equation