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Effective equidistribution of Galois orbits for mildly regular test functions

Emanuel Carneiro, Mithun Kumar Das

TL;DR

This work studies effective versions of Bilu's equidistribution for Galois orbits of small-height points in the $N$-dimensional torus, connecting convergence to the regularity of mildly regular test functions. It develops a Fourier-analytic framework that extends prior work and yields quantitative bounds on the discrepancy $oldsymbol{\\mathcal{E}}(F,oldsymbol{\xi})$ in terms of the Weil height $h(oldsymbol{\xi})$ and the generalized height $h_{oldsymbol{D}}(oldsymbol{\xi})$, under two complementary regularity regimes: (i) regularity on the Fourier side via a class $oldsymbol{\mathcal{A}}$ and (ii) angular regularity at the log-radius via a modulus of continuity or Hölder control. The main results (Theorems 1 and 5, with Corollaries 3 and 6) establish sharp qualitative dependencies, including Hölder exponents up to $\gamma\le 1/2$, and show how angular regularity can yield comparable gains to full Lipschitz regularity in the multi-dimensional setting. The paper also provides sharpness results, and appendices connect the framework to multidimensional angular discrepancy and complete a density-based proof of Bilu's theorem for mildly regular test functions, highlighting the practical impact on discrepancy bounds and equidistribution in higher dimensions.

Abstract

In this paper we provide a detailed study on effective versions of the celebrated Bilu's equidistribution theorem for Galois orbits of sequences of points of small height in the $N$-dimensional algebraic torus, identifying the qualitative dependence of the convergence in terms of the regularity of the test functions considered. We develop a general Fourier analysis framework that extends previous results obtained by Petsche (2005), and by D'Andrea, Narváez-Clauss and Sombra (2017).

Effective equidistribution of Galois orbits for mildly regular test functions

TL;DR

This work studies effective versions of Bilu's equidistribution for Galois orbits of small-height points in the -dimensional torus, connecting convergence to the regularity of mildly regular test functions. It develops a Fourier-analytic framework that extends prior work and yields quantitative bounds on the discrepancy in terms of the Weil height and the generalized height , under two complementary regularity regimes: (i) regularity on the Fourier side via a class and (ii) angular regularity at the log-radius via a modulus of continuity or Hölder control. The main results (Theorems 1 and 5, with Corollaries 3 and 6) establish sharp qualitative dependencies, including Hölder exponents up to , and show how angular regularity can yield comparable gains to full Lipschitz regularity in the multi-dimensional setting. The paper also provides sharpness results, and appendices connect the framework to multidimensional angular discrepancy and complete a density-based proof of Bilu's theorem for mildly regular test functions, highlighting the practical impact on discrepancy bounds and equidistribution in higher dimensions.

Abstract

In this paper we provide a detailed study on effective versions of the celebrated Bilu's equidistribution theorem for Galois orbits of sequences of points of small height in the -dimensional algebraic torus, identifying the qualitative dependence of the convergence in terms of the regularity of the test functions considered. We develop a general Fourier analysis framework that extends previous results obtained by Petsche (2005), and by D'Andrea, Narváez-Clauss and Sombra (2017).

Paper Structure

This paper contains 26 sections, 13 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Effective equidistribution of Galois orbits
  4. Regularity of $F$ on the Fourier space
  5. Angular regularity of $F$ on the Fourier space
  6. Appendices
  7. A word on notation
  8. Consequences of Siegel's lemma
  9. Proofs of Theorems \ref{['Thm1']} and \ref{['Thm4']}
  10. Proof of Theorem \ref{['Thm1']}
  11. Setup
  12. Estimate for $I_1$
  13. Estimate for $I_2$
  14. Proof of Theorem \ref{['Thm4']}
  15. Proofs of Theorems \ref{['Thm5']} and \ref{['Thm7']}
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\{{\bm \xi}_k\}_{k \geq 1} \subset (\overline{\mathbb{Q}}^{\times})^N$ be a strict sequence with $\lim_{k \to \infty} h({\bm \xi}_k) = 0$. Then, for any bounded and continuous function $F:(\mathbb{C}^{\times})^N \to \mathbb{C}$,

Theorems & Definitions (19)

  • Theorem 1: Bilu
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Theorem 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 9 more