A Fourier-Analytic Approach for the Discrepancy of Random Set Systems

TL;DR

This paper introduces a Fourier-analytic method to bound the discrepancy of random set systems, achieving discrepancy at most 1 with high probability under certain conditions, advancing understanding of the Beck-Fiala conjecture.

Contribution

The paper presents a novel Fourier-analytic approach to discrepancy, providing improved bounds for random set systems and offering new techniques potentially applicable to longstanding conjectures.

Findings

Discrepancy at most 1 with high probability when n ≥ Θ(m^2 log m)

New Fourier-based bounds surpass previous results under certain regimes

Method applicable to random set systems with elements included independently

Abstract

One of the prominent open problems in combinatorics is the discrepancy of set systems where each element lies in at most $t$ sets. The Beck-Fiala conjecture suggests that the right bound is $O(\sqrt{t})$, but for three decades the only known bound not depending on the size of the set system has been $O(t)$. Arguably we currently lack techniques for breaking that barrier. In this paper we introduce discrepancy bounds based on Fourier analysis. We demonstrate our method on random set systems. Suppose one has $n$ elements and $m$ sets containing each element independently with probability $p$. We prove that in the regime of $n \geq \Theta(m^2\log(m))$, the discrepancy is at most $1$ with high probability. Previously, a result of Ezra and Lovett gave a bound of $O(1)$ under the stricter assumption that $n \gg m^t$.