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Areal Weil Heights

Preston Kelley

TL;DR

The paper develops areal Weil heights by integrating areal data at a finite set of places into the Favre–Rivera-Letelier adelic height framework. It provides an explicit height formula for h_{ρ_r}, analyzes its relation to the classical Weil height via inequalities, and proves Kronecker-type results that distinguish when small points occur, as well as equidistribution theorems in both subcritical and supercritical capacity regimes. Through the Arakelov-Zhang pairing, the work connects areal heights to auxiliary λ-height systems, enabling comparisons and yielding precise limiting distributions and essential minima. The study culminates with detailed examples, including Lehmer-type phenomena, the arithmeticity threshold for r, and optimal pairings with Chebyshev-type measures, illustrating the rich structure of areal heights in the adelic setting.

Abstract

In 2008, Pritsker introduced the areal Mahler measure, which is defined using an integral over the unit disk, as opposed to the classical Mahler measure which is defined using an integral over the unit circle. In this paper we introduce areal Weil heights, which generalize the areal Mahler measure to the adelic setting. We use the framework of adelic heights established by Favre and Rivera-Letelier and we construct a $p$-adic analog for the area measure on a disk in $\mathbb{C}$. For areal Weil heights we prove an analog of Kronecker's theorem, which characterizes their small points and essential minima. Furthermore, we determine equidistribution theorems for areal Weil heights. In some cases, they have a unique limiting distribution for small points, while in others there are infinitely many limiting distributions. We conclude with examples. In one of our examples, we determine for which radii $r$ there exist sequences of conjugate sets of algebraic integers which uniformly distribute to the disk $D(0,r) \subset \mathbb{C}$, and we compute the limiting height for such sequences.

Areal Weil Heights

TL;DR

The paper develops areal Weil heights by integrating areal data at a finite set of places into the Favre–Rivera-Letelier adelic height framework. It provides an explicit height formula for h_{ρ_r}, analyzes its relation to the classical Weil height via inequalities, and proves Kronecker-type results that distinguish when small points occur, as well as equidistribution theorems in both subcritical and supercritical capacity regimes. Through the Arakelov-Zhang pairing, the work connects areal heights to auxiliary λ-height systems, enabling comparisons and yielding precise limiting distributions and essential minima. The study culminates with detailed examples, including Lehmer-type phenomena, the arithmeticity threshold for r, and optimal pairings with Chebyshev-type measures, illustrating the rich structure of areal heights in the adelic setting.

Abstract

In 2008, Pritsker introduced the areal Mahler measure, which is defined using an integral over the unit disk, as opposed to the classical Mahler measure which is defined using an integral over the unit circle. In this paper we introduce areal Weil heights, which generalize the areal Mahler measure to the adelic setting. We use the framework of adelic heights established by Favre and Rivera-Letelier and we construct a -adic analog for the area measure on a disk in . For areal Weil heights we prove an analog of Kronecker's theorem, which characterizes their small points and essential minima. Furthermore, we determine equidistribution theorems for areal Weil heights. In some cases, they have a unique limiting distribution for small points, while in others there are infinitely many limiting distributions. We conclude with examples. In one of our examples, we determine for which radii there exist sequences of conjugate sets of algebraic integers which uniformly distribute to the disk , and we compute the limiting height for such sequences.

Paper Structure

This paper contains 23 sections, 25 theorems, 173 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Results and Outline of Article
  4. Preliminaries
  5. Basic Notions
  6. Adelic Measures and Adelic Heights
  7. The Arakaelov-Zhang Pairing
  8. gamma r as Adelic Capacity
  9. Definition of Areal Weil Heights
  10. Measure Theory
  11. The Areal Adelic Measure at a Finite Place
  12. Potentials and Energy
  13. The Areal Weil Height
  14. Inequalities
  15. Basic Inequalities
  16. ...and 8 more sections

Key Result

Theorem 1

Let $K$ be a number field, let $S \subset M_K$ be a nonempty finite set of places of $K$, and let $\mathbf{r} = (r_v)_{v \in S} \in (0,\infty)^S$. Then $\rho_\mathbf{r}$ is an adelic measure in the sense of Favre and Rivera-Letelier frl.2. Furthermore, if $\alpha \in L$, where $L/K$ is a finite ext where

Figures (1)

  • Figure 1: Graphs of $y=f_R (\left\lvert x\right\rvert)$ for various $R$. The points $(\pm R, f_R (\left\lvert R\right\rvert))$ are indicated in each graph. In Figure \ref{['graph with radius e^.5']} the graph's intersection with the origin is indicated.

Theorems & Definitions (67)

  • Conjecture 1: Lehmer's Conjecture
  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Proposition 2
  • Example 3
  • Theorem 4
  • Corollary 4
  • Definition 5
  • ...and 57 more