Discrepancy of arithmetic progressions in grids

TL;DR

This paper establishes tight bounds on the discrepancy of arithmetic progressions in multi-dimensional grids, extending classical results and removing previous logarithmic factors, thus advancing understanding in combinatorial discrepancy theory.

Contribution

It provides the first tight bounds for discrepancy in multi-dimensional grids, generalizing the one-dimensional case and improving upon decades-old bounds.

Findings

Discrepancy in $d$-dimensional grids is within a constant factor of $N^{d/(2d+2)}$.

Extended discrepancy bounds to grids with varying side lengths.

Removed polylogarithmic factors from previous upper bounds.

Abstract

We prove that the the discrepancy of arithmetic progressions in the $d$-dimensional grid $\{1, \dots, N\}^d$ is within a constant factor depending only on $d$ of $N^{\frac{d}{2d+2}}$. This extends the case $d=1$, which is a celebrated result of Roth and of Matou\v{s}ek and Spencer, and removes the polylogarithmic factor from the previous upper bound of Valk\'o from about two decades ago. We further prove similarly tight bounds for grids of differing side lengths in many cases.