论文信息 - Exceptional Units and Numbers of Small Mahler Measure

Exceptional Units and Numbers of Small Mahler Measure

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 ).

Joseph H. Silverman

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