On Shallow Packings and Tusn\'ady's Problem

TL;DR

This paper improves bounds on the discrepancy of points and axis-parallel boxes in high-dimensional spaces, settling Tusnádys problem for dimensions five and above, and providing new bounds for lower dimensions.

Contribution

It introduces a novel decomposition technique using shallow cell complexity and extends discrepancy minimization algorithms, achieving optimal bounds in high dimensions.

Findings

For d≥5, bounds match the lower bound of Ω(log^{d-1} n).

Improved bounds for d=2,3,4 match or surpass previous non-constructive bounds.

Provides bounds for discrepancy of points and polytopes, and for anchored boxes with arbitrary measures.

Abstract

Tusn\'ady's problem asks to bound the discrepancy of points and axis-parallel boxes in $\mathbb{R}^d$. Algorithmic bounds on Tusn\'ady's problem use a canonical decomposition of Matou\v{s}ek for the system of points and axis-parallel boxes, together with other techniques like partial coloring and / or random-walk based methods. We use the notion of \emph{shallow cell complexity} and the \emph{shallow packing lemma}, together with the chaining technique, to obtain an improved decomposition of the set system. Coupled with an algorithmic technique of Bansal and Garg for discrepancy minimization, which we also slightly extend, this yields improved algorithmic bounds on Tusn\'ady's problem. For $d\geq 5$, our bound matches the lower bound of $\Omega(\log^{d-1}n)$ given by Matou\v{s}ek, Nikolov and Talwar [IMRN, 2020] -- settling Tusn\'ady's problem, upto constant factors. For $d=2,3,4$, we obtain improved algorithmic bounds of $O(\log^{7/4}n)$, $O(\log^{5/2}n)$ and $O(\log^{13/4}n)$ respectively, which match or improve upon the non-constructive bounds of Nikolov for $d\geq 3$. Further, we also give improved bounds for the discrepancy of set systems of points and polytopes in $\mathbb{R}^d$ generated via translations of a fixed set of hyperplanes. As an application, we also get a bound for the geometric discrepancy of anchored boxes in $\mathbb{R}^d$ with respect to an arbitrary measure, matching the upper bound for the Lebesgue measure, which improves on a result of Aistleitner, Bilyk, and Nikolov [MC and QMC methods, \emph{Springer, Proc. Math. Stat.}, 2018] for $d\geq 4$.