Cubic Goldreich-Levin
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In this paper, we give a cubic Goldreich-Levin algorithm which makes polynomially-many queries to a function f π½_p^n ββ and produces a decomposition of f as a sum of cubic phases and a small error term. This is a natural higher-order generalization of the classical Goldreich-Levin algorithm. The classical (linear) Goldreich-Levin algorithm has wide-ranging applications in learning theory, coding theory and the construction of pseudorandom generators in cryptography, as well as being closely related to Fourier analysis. Higher-order Goldreich-Levin algorithms on the other hand involve central problems in higher-order Fourier analysis, namely the inverse theory of the Gowers U^k norms, which are well-studied in additive combinatorics. The only known result in this direction prior to this work is the quadratic Goldreich-Levin theorem, proved by Tulsiani and Wolf in 2011. The main step of their result involves an algorithmic version of the U^3 inverse theorem. More complications appear in the inverse theory of the U^4 and higher norms. Our cubic Goldreich-Levin algorithm is based on algorithmizing recent work by Gowers and MiliΔeviΔ who proved new quantitative bounds for the U^4 inverse theorem. Our cubic Goldreich-Levin algorithm is constructed from two main tools: an algorithmic U^4 inverse theorem and an arithmetic decomposition result in the style of the Frieze-Kannan graph regularity lemma. As one application of our main theorem we solve the problem of self-correction for cubic Reed-Muller codes beyond the list decoding radius. Additionally we give a purely combinatorial result: an improvement of the quantitative bounds on the U^4 inverse theorem.
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