Sparse Geometric Set Systems and the Beck-Fiala Conjecture

TL;DR

This paper establishes new discrepancy bounds for geometric set systems with bounded shallow cell complexity, verifying and improving the Beck-Fiala conjecture in several geometric cases using matchings with low crossing numbers.

Contribution

It introduces discrepancy bounds for geometric set systems with shallow cell complexity, utilizing matchings with low crossing numbers and advanced combinatorial techniques.

Findings

Discrepancy bounds are $o(\sqrt{t})$ for certain geometric set systems.

Existence of matchings with low crossing number is demonstrated.

Improves upon the Beck-Fiala conjecture for specific geometric configurations.

Abstract

We investigate the combinatorial discrepancy of geometric set systems having bounded shallow cell complexity in the \emph{Beck-Fiala} setting, where each point belongs to at most $t$ ranges. For set systems with shallow cell complexity $\psi(m,k)=g(m)k^{c}$, where $(i)$ $g(m) = o(m^{\varepsilon})$ for any $\varepsilon\in (0,1],$ $(ii)$ $\psi$ is non-decreasing in $m$, and $(iii)$ $c>0$ is independent of $m$ and $k$, we get a discrepancy bound of \[ O\left(\sqrt{\left(\log n+\left(t^{c}g(n)\right)^{\frac{1}{1+c}}\right)\log n}\right).\] For $t=\omega(\log^2 n)$, in several cases, such as for set systems of points and half-planes / disks / pseudo-disks in $\mathbb{R}^2$, points and orthants in $\mathbb{R}^3$ etc., these bounds are $o(\sqrt{t})$, which verifies (and improves upon) the conjectured bound of Beck and Fiala~\emph{(Disc. Appl. Math., 1981)}. Our bounds are obtained by showing the existence of \emph{matchings with low crossing number}, using the multiplicative weights update method of Welzl \emph{(SoCG, 1988)}, together with the recent bound of Mustafa \emph{(Disc. Comp. Geom., 2015)} on \emph{shallow packings} of set systems in terms of their shallow cell complexity. For set systems of shallow cell complexity $\psi(m,k)=m^{c_1}g(m)k^{c}$, we obtain matchings with crossing number at most \[ O\left(\left(n^{c_1}g(n)t^{c}\right)^{\frac{1}{1+c_1+c}}\right).\] These are of independent interest.