The exceptional set in Cassel's theorem on small cyclotomic integers

TL;DR

This paper classifies small cyclotomic algebraic integers with a house less than 2, completing a 1965 conjecture by analyzing the exceptional set identified by Cassels and building on prior work.

Contribution

It precisely determines the exceptional set of cyclotomic integers with small house, resolving Robinson's 1965 conjecture for house less than 2.

Findings

Determined the exceptional set for house at most √5.

Resolved Robinson's conjecture for house less than 2.

Extended previous classifications with new explicit results.

Abstract

In a 1965 paper, R. Robinson made five conjectures about the classification of cyclotomic algebraic integers for which the maximum absolute value in any complex embedding (the house) is small, modulo the equivalence relation generated by Galois conjugation and multiplication by roots of unity. In response to one of these conjectures, Cassels showed in 1969 that when the house is at most $\sqrt{5}$, one obtains three parametric families plus an effectively computable finite set of equivalence classes of exceptions. Building on the work of Jones, Calegari-Morrison-Snyder, and Robinson-Wurtz, we determine this exceptional set. By specializing to the case where the house is strictly less than 2, we resolve the final outstanding conjecture from Robinson's 1965 paper.