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Around Fekete's theorem

Norm Levenberg, Mayuresh Londhe

TL;DR

The paper investigates when a compact set $\Sigma$ in the complex plane can contain infinitely many conjugate algebraic integers by leveraging potential theory, heights relative to $\Sigma$, and Mahler measures. It constructs sequences whose heights approach $-\log c(\Sigma)$ and whose conjugates concentrate near $\Sigma$, with limiting distributions linked to the equilibrium measure $\mu_{\Sigma}$ in favorable cases. A general framework is developed for limiting measures: any subsequential limit $\mu$ of conjugates with bounded $h_{\Sigma}$ satisfies an arithmetic inequality involving polynomials, the leading coefficients, and an escape rate, and has finite energy. These results connect Fekete-type phenomena to equilibrium theory and raise questions about tightening bounds, the universality of limiting distributions, and arithmetic probability measures on $\Sigma$.

Abstract

A classical result of Fekete gives necessary conditions on a compact set in the complex plane so that it contains infinitely many sets of conjugate algebraic integers. For such sets, we demonstrate the existence of a sequence of algebraic integers such that most of their conjugates eventually lie near the set, while maintaining a bound on heights. Finally, we examine properties satisfied by the limiting distribution of a sequence of algebraic numbers.

Around Fekete's theorem

TL;DR

The paper investigates when a compact set in the complex plane can contain infinitely many conjugate algebraic integers by leveraging potential theory, heights relative to , and Mahler measures. It constructs sequences whose heights approach and whose conjugates concentrate near , with limiting distributions linked to the equilibrium measure in favorable cases. A general framework is developed for limiting measures: any subsequential limit of conjugates with bounded satisfies an arithmetic inequality involving polynomials, the leading coefficients, and an escape rate, and has finite energy. These results connect Fekete-type phenomena to equilibrium theory and raise questions about tightening bounds, the universality of limiting distributions, and arithmetic probability measures on .

Abstract

A classical result of Fekete gives necessary conditions on a compact set in the complex plane so that it contains infinitely many sets of conjugate algebraic integers. For such sets, we demonstrate the existence of a sequence of algebraic integers such that most of their conjugates eventually lie near the set, while maintaining a bound on heights. Finally, we examine properties satisfied by the limiting distribution of a sequence of algebraic numbers.
Paper Structure (7 sections, 8 theorems, 89 equations)

This paper contains 7 sections, 8 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Potential theory
  4. Heights and Mahler measure
  5. Proof of Theorem \ref{['T:existence']}
  6. Proof of Theorem \ref{['T:limiting']}
  7. Further discussion

Key Result

Theorem 1.3

Let $\Sigma \subset \mathbb{C}$ be compact and symmetric about the real axis. If $0< {c}(\Sigma) <1$, then there exists a sequence $\{\alpha_n\}$ of distinct algebraic integers such that $h_{\Sigma}(\alpha_n) \to - \log {{c}(\Sigma)}$ as $n \to \infty$, and for any neighborhood $U$ of $\Sigma$, we h where $\sharp \{{\rm {Conj}} (\alpha_n) \cap U\}$ denotes the number of conjugates of $\alpha_n$ co

Theorems & Definitions (16)

  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Remark 3.1
  • Definition 4.2
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • ...and 6 more