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.

在此调查中，我们探索序列\（f_1，f_2，f_3，\ ldots \）中的函数之间的超正交性如何导致相关平方函数的正不等式或逆不等式。我们区分了三种主要的超正交性类型，我们证明了它们在谐波分析和数论的多种设置中出现。这种观点为中心结果提供了清晰的证明，并统一了包括Khintchine不等式，Walsh–Paley级数，离散算子，解耦，丢番亭方程组的计数解，跟踪函数的多重相关性以及针对短字符和的Burgess主题。