# Seminar: November 22

## Madhur Tulsiani, TTIC

## Quadratic Goldreich-Levin Theorems

Decomposition theorems in classical Fourier analysis enable us to express a bounded function in terms of few linear phases with large Fourier coefficients plus a part that is pseudorandom with respect to linear phases. The Goldreich-Levin algorithm can be viewed as an algorithmic analogue of such a decomposition as it gives a way to efficiently find the linear phases associated with large Fourier coefficients.

In the study of "quadratic Fourier analysis", higher-degree analogues of such decompositions have been developed in which the pseudorandomness property is stronger but the structured part correspondingly weaker. For example, it has previously been shown that it is possible to express a bounded function as a sum of a few quadratic phases plus a part that is small in the U_3 norm, defined by Gowers for the purpose of counting arithmetic progressions of length 4. We give a polynomial time algorithm for computing such a decomposition.

A key part of the algorithm is a local self-correction procedure for Reed-Muller codes of order 2 for a function at distance 1/2-\epsilon from a codeword. Given a Boolean function on n bits at fractional Hamming distance 1/2 - \epsilon from a quadratic phase (which is a codeword of Reed- Muller code of order 2), we give an algorithm that runs in time polynomial in n and finds a codeword at distance at most 1/2 - \eta for \eta = \eta(\epsilon). This is an algorithmic analogue of Samorodnitsky's result], which gave a tester for the above problem. In the process, we give algorithmic versions of results from additive combinatorics used in Samorodnitsky's proof and a refined version of the inverse theorem for the Gowers U3 norm over F_2^n .

Joint work with Julia Wolf.