Tight Bounds for Tusnády's Problem in the Plane

2026-10-06

AI Summary

This study investigates the tight worst-case bounds on the combinatorial discrepancy of axis-parallel rectangles in the plane. For a set of n points, we present a concise, elementary proof achieving matching lower bounds by combining coordinate-wise analysis, the accumulation of oscillating potential functions, and bounded difference control via the union bound method. Our results establish that this combinatorial discrepancy is Θ(log^{3/2} n), thereby closing a long-standing gap between known upper and lower bounds. Furthermore, we demonstrate that with extremely high probability, there exist anchored rectangles exhibiting an imbalance of at least c(log n)^{3/2}. These findings reveal the theoretical limits of optimal imbalance for random point sets and confirm that the previously known upper bound is indeed tight, resolving a longstanding open problem in the field.

Abstract

We show that the worst-case combinatorial discrepancy of $n$ points in the plane with respect to axis-parallel rectangles is $Θ(\log^{3/2}n)$. The known bounds were $Ω(\log n)$ and $O(\log^{3/2}n)$; we prove the matching lower bound. It holds for random point sets: for every $A>0$, there is a constant $c_A>0$ such that, with probability at least $1-e^{-An}$, every coloring of $n$ independent uniform points in the unit square has an anchored rectangle with imbalance at least $c_A(\log_2n)^{3/2}$. The proof is surprisingly simple and elementary. It reveals one coordinate digit by digit. With overwhelming probability over the points, the conditional gains of an oscillation potential add up to $Ω(\log^{3/2}n)$ over $Θ(\log n)$ digits. Bounded differences control the fluctuations well enough for a union bound over all colorings.

Problem

• Tusnády's problem
• combinatorial discrepancy
• axis-parallel rectangles
• tight bounds
• random point sets

Innovation

• Combinatorial Discrepancy
• Tusnády's Problem
• Oscillation Potential
• Bounded Differences
• Axis-parallel Rectangles