Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a discrete approach to lower bounds in discrepancy theory

Luca Brandolini, Bianca Gariboldi, Giacomo Gigante, Alessandro Monguzzi

TL;DR

The paper develops discrete analogues of classical discrepancy lower bounds by restricting to finite, grid-based families of geometric ranges on the torus, including anchored boxes, cubes, and balls. Using a $b$-adic Haar basis and discrete Fourier techniques, it proves Roth-type lower bounds in the $L^2$-sense over a grid with size $M=b^{\nu+\tau}$ tied to the point-set size $N$, namely $\|D_N\|_2 \ge c\big(\log_b N\big)^{(d-1)/2}$ (with a stronger endpoint bound $\|D_N\|_\infty \ge c\log N$ in dimension 2). It then extends these discrete lower bounds to cubes (squares) and balls, providing explicit grid-scale conditions (e.g., $M\ge 18dN$ for cubes; $M\ge C N^{1+1/(2d)} r^{-1}$ for balls) that guarantee nontrivial irregularity against these finite families. Overall, the work uncovers how nontrivial irregularity persists for finite collections of sets and unifies continuous discrepancy theory with discretized, grid-driven analyses useful for applications in numerical integration and point distribution.

Abstract

In this paper, we prove that some renowned lower bounds in discrepancy theory admit a discrete analogue. Namely, we prove that the lower bound of the discrepancy for corners in the unit cube due to Roth holds true also for a suitable finite family of corners. We also prove two analogous results for the discrepancy on the torus with respect to squares and balls.

On a discrete approach to lower bounds in discrepancy theory

TL;DR

The paper develops discrete analogues of classical discrepancy lower bounds by restricting to finite, grid-based families of geometric ranges on the torus, including anchored boxes, cubes, and balls. Using a -adic Haar basis and discrete Fourier techniques, it proves Roth-type lower bounds in the -sense over a grid with size tied to the point-set size , namely (with a stronger endpoint bound in dimension 2). It then extends these discrete lower bounds to cubes (squares) and balls, providing explicit grid-scale conditions (e.g., for cubes; for balls) that guarantee nontrivial irregularity against these finite families. Overall, the work uncovers how nontrivial irregularity persists for finite collections of sets and unifies continuous discrepancy theory with discretized, grid-driven analyses useful for applications in numerical integration and point distribution.

Abstract

In this paper, we prove that some renowned lower bounds in discrepancy theory admit a discrete analogue. Namely, we prove that the lower bound of the discrepancy for corners in the unit cube due to Roth holds true also for a suitable finite family of corners. We also prove two analogous results for the discrepancy on the torus with respect to squares and balls.
Paper Structure (4 sections, 14 theorems, 156 equations, 1 figure)

This paper contains 4 sections, 14 theorems, 156 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of the discrete version of Roth's lower bound
  3. Proof of the lower bound for squares
  4. Proof of the lower bound for balls

Key Result

Theorem 1

Let $b$ be a positive integer greater than $1$. Let $N$ be a positive integer, let $\nu$ be the integer such that $b^{\nu-2}\leqslant N<b^{\nu-1}$, and let $M=b^{\nu+\tau}$ for an integer $\tau\ge 1$. Then there exists an explicit constant $c>0$, depending only on $b$ and $d$, such that, for every f In particular, there exists an anchored box $A(j_1/M, \cdots, j_d/M)$ such that When $d=2$ there e

Figures (1)

  • Figure 1: The counting term of the discrepancy function.

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm: roth-discreto']}
  • Remark 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 8
  • ...and 19 more