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

2025/10/23 by Bajpai, Jitendra, Das, Srijan, Kedlaya, Kiran S. +4
#11Y40 #FOS: Mathematics #Number Theory (math.NT) #Primary 11R18 #secondary 11R06

paper · doi:10.48550/arxiv.2510.20435

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 √(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.

Citations

• Space vectors forming rational angles
• The Cassels heights of cyclotomic integers
• A proof of the Schinzel-Zassenhaus conjecture on polynomials
• On the magnitudes of some small cyclotomic integers

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