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16
The inverse conjecture for the Gowers norm over finite fields via the correspondence principle
, 2010
"... The inverse conjecture for the Gowers norms U d(V) for finitedimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ‖ f ‖Ud (V) if and only if it correlates with a phase polynomial φ = eF(P) of degree at most d−1, thus P:V→F is a ..."
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Cited by 39 (8 self)
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The inverse conjecture for the Gowers norms U d(V) for finitedimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ‖ f ‖Ud (V) if and only if it correlates with a phase polynomial φ = eF(P) of degree at most d−1, thus P:V→F is a polynomial of degree at most d − 1. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case char F> d from an ergodic theory counterpart, which was recently established by Bergelson, Tao and Ziegler. In low characteristic we obtain a partial result, in which the phase polynomial φ is allowed to be of some larger degree C(d). The full inverse conjecture remains open in low characteristic; the counterexamples found so far in this setting can be avoided by a slight reformulation of the conjecture.
On the Bogolyubov–Ruzsa lemma
, 2012
"... Our main result is that if A is a finite subset of an abelian group with jACAj 6 KjAj, then 2A 2A contains an O.logO.1 / 2K/dimensional coset progression M of size at least exp.O.logO.1 / 2K//jAj. ..."
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Cited by 11 (1 self)
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Our main result is that if A is a finite subset of an abelian group with jACAj 6 KjAj, then 2A 2A contains an O.logO.1 / 2K/dimensional coset progression M of size at least exp.O.logO.1 / 2K//jAj.
Quadratic GoldreichLevin theorems
 In Proc. 52th Annu
"... 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 GoldreichLevin algorithm [GL89] can be viewed as an algorithmic analogue of ..."
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Cited by 5 (1 self)
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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 GoldreichLevin algorithm [GL89] 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”, higherdegree 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 selfcorrection procedure for ReedMuller codes of order 2 (over Fn 2) for a function at distance 1/2−ε from a codeword. Given a function f: Fn 2 → {−1, 1} at fractional Hamming distance 1/2 − ε from a quadratic phase (which is a codeword of ReedMuller 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 − η for η = η(ε). This is an algorithmic analogue of Samorodnitsky’s result [Sam07], which gave a tester for the above problem. To our knowledge, it represents the first instance of a correction procedure for any class of codes, beyond the listdecoding radius. 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 U 3 norm
Higher order Fourier analysis of multiplicative functions and applications
, 2014
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Uniformity of multiplicative functions and partition regularity of some quadratic equations
, 2013
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Advice Lower Bounds for the Dense Model Theorem
, 2011
"... We prove a lower bound on the amount of nonuniform advice needed by blackbox reductions for the Dense Model Theorem of Green, Tao, and Ziegler, and of Reingold, Trevisan, Tulsiani, and Vadhan. The latter theorem roughly says that for every distribution D that is δdense in a distribution that is ǫ ..."
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Cited by 1 (1 self)
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We prove a lower bound on the amount of nonuniform advice needed by blackbox reductions for the Dense Model Theorem of Green, Tao, and Ziegler, and of Reingold, Trevisan, Tulsiani, and Vadhan. The latter theorem roughly says that for every distribution D that is δdense in a distribution that is ǫ ′indistinguishable from uniform, there exists a “dense model ” for D, that is, a distribution that is δdense in the uniform distribution and is ǫindistinguishable from D. This ǫindistinguishability is with respect to an arbitrary small class of functions F. For the very natural case where ǫ ′ ≥ Ω(ǫδ) and ǫ ≥ δ O(1) , our lower bound implies that Ω ( √ (1/ǫ)log(1/δ) · log F  ) advice bits are necessary. There is only a polynomial gap between our lower bound and the best upper bound for this case (due to Zhang), which is O ( (1/ǫ 2)log(1/δ)·log F  ). Our lower bound can be viewed as an analog of list size lower bounds for listdecoding of errorcorrecting codes, but for “dense model decoding ” instead. 1