On the magnitudes of some small cyclotomic integers
Volume 160 / 2013
Acta Arithmetica 160 (2013), 317-332
MSC: Primary 11R18.
DOI: 10.4064/aa160-4-2
Abstract
We prove the last of five outstanding conjectures made by R. M. Robinson from 1965 concerning small cyclotomic integers. In particular, given any cyclotomic integer $\beta $ all of whose conjugates have absolute value at most $5$, we prove that the largest such conjugate has absolute value of one of four explicit types given by two infinite classes and two exceptional cases. We also extend this result by showing that with the addition of one form, the conjecture is true for $\beta $ with magnitudes up to $5+ {1/25}$.
