An elementary proof of a lower bound for the inverse of the star discrepancy

TL;DR

This paper provides an elementary proof establishing a lower bound on the number of points needed for nearly uniform distribution in high-dimensional unit cubes, improving understanding of discrepancy theory.

Contribution

It offers a simplified proof of the best known lower bound for the inverse star discrepancy, connecting point count, dimension, and accuracy.

Findings

Proves that the number of points n must grow at least linearly with dimension d and inversely with error ε.

Simplifies the proof of a key result in discrepancy theory.

Confirms the lower bound n ≳ d · ε^{-1} for point distributions in [0,1]^d.

Abstract

A central problem in discrepancy theory is the challenge of evenly distributing points $\left\{x_1, \dots, x_n \right\}$ in $[0,1]^d$. Suppose a set is so regular that for some $\varepsilon> 0$ and all $y \in [0,1]^d$ the sub-region $[0,y] = [0,y_1] \times \dots \times [0,y_d]$ contains a number of points nearly proportional to its volume and $$\forall~y \in [0,1]^d \qquad \left| \frac{1}{n} \# \left\{1 \leq i \leq n: x_i \in [0,y] \right\} - \mbox{vol}([0,y]) \right| \leq \varepsilon,$$ how large does $n$ have to be depending on $d$ and $\varepsilon$? We give an elementary proof of the currently best known result, due to Hinrichs, showing that $n \gtrsim d \cdot \varepsilon^{-1}$.