On Salem numbers which are exceptional units

TL;DR

This paper extends previous constructions to show that for various degrees and integers, there are infinitely many Salem numbers with the property that their n-th power minus one is a unit, revealing new algebraic number theory phenomena.

Contribution

It introduces new infinite families of Salem numbers with units derived from powers minus one, generalizing earlier results and expanding understanding of Salem number properties.

Findings

Existence of infinitely many Salem numbers with specified algebraic properties

Construction methods for Salem numbers with units from powers minus one

Generalization of previous results to broader classes of degrees and integers

Abstract

By extending a construction due to Gross and McMullen [2], we show that for any odd integer n and for any even integer d>n+2 there are infinitely many Salem numbers $\alpha$ of degree d such that $\alpha^n-1$ is a unit. A similar result is also proved when n runs through some classes of even integers, d>n+3 and d/2 is an odd integer.