An improvement of the Beck-Fiala theorem

TL;DR

This paper improves the upper bound on the discrepancy of set systems with degree d, refining the previous bounds from 2d-2 to 2d-log* d, advancing the theoretical understanding of discrepancy theory.

Contribution

The paper presents a tighter upper bound for the discrepancy of set systems of degree d, reducing it from 2d-4 to 2d-log* d, which is a significant theoretical improvement.

Findings

Bound reduced from 2d-4 to 2d-log* d

Advances the theoretical understanding of discrepancy bounds

Improves previous bounds established in 1981

Abstract

In 1981 Beck and Fiala proved an upper bound for the discrepancy of a set system of degree d that is independent of the size of the ground set. In the intervening years the bound has been decreased from 2d-2 to 2d-4. We improve the bound to 2d-log* d.