Fourier growth of structured 𝔽_2-polynomials and applications
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We analyze the Fourier growth, i.e. the L_1 Fourier weight at level k (denoted L_1,k), of various well-studied classes of "structured" 𝔽_2-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [CHHL19,CHLT19,CGLSS20] which show that upper bounds on Fourier growth (even at level k=2) give unconditional pseudorandom generators. Our main structural results on Fourier growth are as follows: - We show that any symmetric degree-d 𝔽_2-polynomial p has L_1,k(p) ≤[p=1] · O(d)^k, and this is tight for any constant k. This quadratically strengthens an earlier bound that was implicit in [RSV13]. - We show that any read-Δ degree-d 𝔽_2-polynomial p has L_1,k(p) ≤[p=1] · (k Δ d)^O(k). - We establish a composition theorem which gives L_1,k bounds on disjoint compositions of functions that are closed under restrictions and admit L_1,k bounds. Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of 𝔽_2-polynomials.
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