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

Jitendra Bajpai, Srijan Das, Kiran S. Kedlaya, Nam H. Le, Meghan Lee, Antoine Leudière, Jorge Mello

TL;DR

The paper completes the Cassels-based program for classifying cyclotomic algebraic integers with small house by pinpointing the exceptional set and confirming that every $ ext{}^2<5.01$ is either one of the Cassels families, equivalent to a simple root-of-unity expression, or represented by an explicit finite list. It refines the Cassels height calculus through minimal-level analysis, $p$-decompositions, and combinatorial constraints, then integrates Artin reciprocity-based prime-splitting arguments to prune possibilities. A three-stage computational strategy (handling higher prime powers, large primes, and small primes) combined with precise arithmetic and a corrected Robinson–Wurtz framework yields a full resolution of Robinson’s conjectures in the $ ext{house}<2$ regime, and corroborates the broader structure of castles for cyclotomic integers. The results provide a concrete, verifiable classification that deepens understanding of small cyclotomic integers and demonstrates the effectiveness of combining height theory, combinatorics, and computational verification in explicit number-theoretic problems.

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.

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

TL;DR

The paper completes the Cassels-based program for classifying cyclotomic algebraic integers with small house by pinpointing the exceptional set and confirming that every is either one of the Cassels families, equivalent to a simple root-of-unity expression, or represented by an explicit finite list. It refines the Cassels height calculus through minimal-level analysis, -decompositions, and combinatorial constraints, then integrates Artin reciprocity-based prime-splitting arguments to prune possibilities. A three-stage computational strategy (handling higher prime powers, large primes, and small primes) combined with precise arithmetic and a corrected Robinson–Wurtz framework yields a full resolution of Robinson’s conjectures in the regime, and corroborates the broader structure of castles for cyclotomic integers. The results provide a concrete, verifiable classification that deepens understanding of small cyclotomic integers and demonstrates the effectiveness of combining height theory, combinatorics, and computational verification in explicit number-theoretic problems.

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 , 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.
Paper Structure (66 sections, 51 theorems, 87 equations, 7 tables, 4 algorithms)

This paper contains 66 sections, 51 theorems, 87 equations, 7 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. General setup
  3. The theorem of Cassels
  4. Additional metrics
  5. Prior intermediate results
  6. Overview of the proof
  7. Future directions
  8. The calculus of Cassels heights
  9. Minimal level
  10. Calculation of some minimal levels
  11. Decomposition and the Cassels height: prime power case
  12. Decomposition and the Cassels height: prime case
  13. Lower bounds on the Cassels height
  14. Computational methods
  15. Computing the basic quantities
  16. ...and 51 more sections

Key Result

Theorem 1.1

For $\alpha$ a cyclotomic algebraic integer, $\hbox{$\alpha$}^2 < 5.01$ if and only if $\alpha$ satisfies oneConditions (1)--(3) are not mutually exclusive, but the overlaps between them are finite and known. For instance, the overlaps between (2) and (3) are the subject of robinson, for which see m

Theorems & Definitions (130)

  • Theorem 1.1: Cassels
  • Theorem 1.2
  • Theorem 1.3: Loxton
  • proof
  • Lemma 1.4: Cassels
  • Theorem 1.5: Jones
  • Theorem 1.6: Calegari--Morrison--Snyder
  • Theorem 1.7: Robinson--Wurtz
  • proof
  • Corollary 1.8: Robinson--Wurtz
  • ...and 120 more