2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/116706This 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.73 pagesClassical Analysis and ODEsNumber Theory11K38, 60G15text