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 = √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.
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BibTeX
@misc{bajpai2026classification,
title = {{On the classification of small cyclotomic integers}},
author = {Jitendra Bajpai and Srijan Das and Kiran S. Kedlaya and Jorge Mello},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.05853},
eprint = {2609.05853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05853},
}
RIS
TY - UNPB
TI - On the classification of small cyclotomic integers
AU - Jitendra Bajpai
AU - Srijan Das
AU - Kiran S. Kedlaya
AU - Jorge Mello
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.05853
UR - https://arxiv.org/abs/2609.05853
AB - 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 = √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.
ER -