On superorthogonality

TL;DR

This survey investigates superorthogonality in function sequences, showing how it leads to inequalities for square functions and unifies various results in harmonic analysis and number theory.

Contribution

It introduces a unified perspective on superorthogonality, providing simplified proofs and connecting diverse topics across harmonic analysis and number theory.

Findings

Identifies three main types of superorthogonality.

Demonstrates superorthogonality's role in key inequalities.

Unifies multiple results in harmonic analysis and number theory.

Abstract

In this survey, we explore how superorthogonality amongst functions in a sequence $f_1,f_2,f_3,\ldots$ 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.