On the classification of small cyclotomic integers

Abstract

We give a general classification theorem for cyclotomic algebraic integers with all complex absolute values bounded by a fixed constant $c$, modeled on the theorem of Cassels which treats the case $c=\sqrt{5}$ up to finitely many exceptions. As a corollary, we establish that the range of the function taking a cyclotomic integer to its maximum complex absolute value is a well-ordered (but not closed) subset of the real numbers. We also formulate analogous statements for algebraic numbers in the maximal cyclotomic extension of a fixed number field. The proofs combine a result of Loxton, which bounds the number of roots of unity in the shortest additive representation of a cyclotomic integer in terms of the maximum complex absolute value, with an equidistribution theorem of Bilu et al. for Galois orbits of torsion points on algebraic tori.