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Irregularities of distribution and Fourier transforms of multi-dimensional convex bodies

Luca Brandolini, Leonardo Colzani, Giancarlo Travaglini

TL;DR

This work extends irregularities-of-distribution results from balls to general convex bodies with smooth boundaries and finite order of contact on the torus. It develops geometric Fourier-transform estimates for the convex body's indicator, notably using parallel-section functions and BNW-type bounds, and then translates these into distribution lower bounds via a Cassels–Montgomery framework. The main contribution is a $c\,N^{1-1/d}$ averaged discrepancy bound for convex bodies, along with a uniform lower bound on the Fourier transform in terms of parallel sections, showing that smooth boundary behavior suffices to match the ball case in many regimes. The results bridge geometric discrepancy for balls and cubes and advance understanding of how boundary regularity influences Fourier decay and lattice-point discrepancy in higher dimensions.

Abstract

W. Schmidt, H. Montgomery, and J. Beck proved a result on irregularities of distribution with respect to $d$-dimensional balls. In this paper, we extend their result to any $d$-dimensional convex body with a smooth boundary and finite order of contact. As an intermediate step, we prove a geometric inequality for the Fourier transform of the characteristic function of a convex body.

Irregularities of distribution and Fourier transforms of multi-dimensional convex bodies

TL;DR

This work extends irregularities-of-distribution results from balls to general convex bodies with smooth boundaries and finite order of contact on the torus. It develops geometric Fourier-transform estimates for the convex body's indicator, notably using parallel-section functions and BNW-type bounds, and then translates these into distribution lower bounds via a Cassels–Montgomery framework. The main contribution is a averaged discrepancy bound for convex bodies, along with a uniform lower bound on the Fourier transform in terms of parallel sections, showing that smooth boundary behavior suffices to match the ball case in many regimes. The results bridge geometric discrepancy for balls and cubes and advance understanding of how boundary regularity influences Fourier decay and lattice-point discrepancy in higher dimensions.

Abstract

W. Schmidt, H. Montgomery, and J. Beck proved a result on irregularities of distribution with respect to -dimensional balls. In this paper, we extend their result to any -dimensional convex body with a smooth boundary and finite order of contact. As an intermediate step, we prove a geometric inequality for the Fourier transform of the characteristic function of a convex body.

Paper Structure

This paper contains 4 sections, 13 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Geometric estimates for the Fourier transform
  3. Proof of Theorem \ref{['easy']} and Theorem \ref{['thm:da sotto']}
  4. Proof of Theorem \ref{['d3']}

Key Result

Theorem 1

Let $d\geqslant2$, and let $\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$ be the $d$-dimensional torus. Let $B\subset \mathbb{T}^{d}$ be a ball of diameter less than $1$. Then there exists a positive constant $c$ such that, for any choice of $N$ points $z_{1},\ldots,z_{N}$ in $\mathbb{T}^{d}$, we ha where $\left\vert B\right\vert$ is the volume of the ball $B$, and $\chi_K(t)$ denotes the characte

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 6
  • Corollary 7
  • proof : Proof of Corollary \ref{['BNW1']}
  • Theorem 8
  • Theorem 9
  • ...and 13 more