The two-dimensional small ball inequality and binary nets

TL;DR

This paper introduces a new proof of the two-dimensional small ball inequality, links it to discrepancy theory through binary nets, and proposes a method for higher-dimensional cases via dimension reduction.

Contribution

It provides the first formal connection between the small ball inequality and binary nets, and outlines a potential approach for higher dimensions.

Findings

New proof of the 2D small ball inequality.

Established connection between small ball inequality and binary nets.

Proposed dimension reduction approach for higher dimensions.

Abstract

In the current paper we present a new proof of the small ball inequality in two dimensions. More importantly, this new argument, based on an approach inspired by lacunary Fourier series, reveals the first formal connection between this inequality and discrepancy theory, namely the construction of two-dimensional binary nets, i.e. finite sets which are perfectly distributed with respect to dyadic rectangles. This relation allows one to generate all possible point distributions of this type. In addition, we outline a potential approach to the higher-dimensional small ball inequality by a dimension reduction argument. In particular this gives yet another proof of the two-dimensional signed (i.e. coefficients $\pm 1$) small ball inequality by reducing it to a simple one-dimensional estimate. However, we show that an analogous estimate fails to hold for arbitrary coefficients.