Small Ball and Discrepancy Inequalities

TL;DR

This paper proves a new inequality for sums of hyperbolic Haar functions in three variables, advancing the Small Ball Conjecture and connecting harmonic analysis, discrepancy theory, and probability.

Contribution

It provides a comprehensive proof of the Small Ball inequality in three variables, extending previous results and including detailed background and related conjectures.

Findings

Proved the Small Ball inequality in three variables.

Connected discrepancy inequalities with harmonic analysis and probability.

Provided detailed background and proof techniques.

Abstract

This is a comprehensive set of notes on the ArXiV paper math.CA/0609815 by Dmitry Bilyk and the author. The focus of that paper is a new inequality for sums of hyperbolic Haar functions in three variables, extending a famous result of J Beck from 1987. This is an improvement on what is known as the Small Ball Conjecture. In this paper, that result is proved, in a more leisurely fashion and additional remarks. In addition, background material is gathered together, including a complete proof of the necessary Harmonic Analysis; a summary of known results on the Small Ball inequality; Irregularities of Distribution; the relationship with conjectures in Approximation Theory and Probability Theory.