On superorthogonality

Abstract

In this survey, we explore how superorthogonality amongst functions in a sequence $f_1,f_2,f_3,\dots$ results in direct or converse inequalities for an associated square function. We distinguish between three main types of superorthogonality, which we demonstrate arise in a wide array of settings in harmonic analysis and number theory. This perspective gives clean proofs of central results, and unifies topics including Khintchine's inequality, Walsh-Paley series, discrete operators, decoupling, counting solutions to systems of Diophantine equations, multicorrelation of trace functions, and the Burgess bound for short character sums.

61 pages. With an appendix by Emmanuel Kowalski. V2 corrects minor typos