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40
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.
The sum of d smallbias generators fools polynomials of degree d
 In IEEE Conference on Computational Complexity
, 2007
"... We prove that the sum of d smallbias generators L: F s → F n fools degreed polynomials in n variables over a prime field F, for any fixed degree d and field F, including F = F2 = {0, 1}. Our result improves on both the work by Bogdanov and Viola (FOCS ’07) and the beautiful followup by Lovett (ST ..."
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Cited by 31 (2 self)
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We prove that the sum of d smallbias generators L: F s → F n fools degreed polynomials in n variables over a prime field F, for any fixed degree d and field F, including F = F2 = {0, 1}. Our result improves on both the work by Bogdanov and Viola (FOCS ’07) and the beautiful followup by Lovett (STOC ’08). The first relies on a conjecture that turned out to be true only for some degrees and fields, while the latter considers the sum of 2 d smallbias generators (as opposed to d in our result). Our proof builds on and somewhat simplifies the arguments by Bogdanov and Viola (FOCS ’07) and by Lovett (STOC ’08). Its core is a case analysis based on the bias of the polynomial to be fooled. 1
ListDecoding ReedMuller codes over small fields
 IN PROC. 40 TH ACM SYMP. ON THEORY OF COMPUTING (STOC’08)
, 2008
"... We present the first local listdecoding algorithm for the r th order ReedMuller code RM(r, m) over F2 for r ≥ 2. Given an oracle for a received word R: F m 2 → F2, our randomized local listdecoding algorithm produces a list containing all degree r polynomials within relative distance (2 −r − ε) f ..."
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Cited by 22 (4 self)
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We present the first local listdecoding algorithm for the r th order ReedMuller code RM(r, m) over F2 for r ≥ 2. Given an oracle for a received word R: F m 2 → F2, our randomized local listdecoding algorithm produces a list containing all degree r polynomials within relative distance (2 −r − ε) from R for any ε> 0 in time poly(m r, ε −r). The list size could be exponential in m at radius 2 −r, so our bound is optimal in the local setting. Since RM(r, m) has relative distance 2 −r, our algorithm beats the Johnson bound for r ≥ 2. In the setting where we are allowed runningtime polynomial in the blocklength, we show that listdecoding is possible up to even larger radii, beyond the minimum distance. We give a deterministic listdecoder that works at error rate below J(2 1−r), where J(δ) denotes the Johnson radius for minimum distance δ. This shows that RM(2, m) codes are listdecodable up to radius η for any constant η < 1 in time 2 polynomial in the blocklength. Over small fields Fq, we present listdecoding algorithms in both the global and local settings that work up to the listdecoding radius. We conjecture that the listdecoding radius approaches the minimum distance (like over F2), and prove this holds true when the degree is divisible by q − 1.
Inverse Conjecture for the Gowers norm is false
 In Proceedings of the 40th Annual ACM Symposium on the Theory of Computing (STOC
, 2007
"... Let p be a fixed prime number, and N be a large integer. The ’Inverse Conjecture for the Gowers norm ’ states that if the ”dth Gowers norm ” of a function f: F N p → F is nonnegligible, that is larger than a constant independent of N, then f can be nontrivially approximated by a degree d − 1 poly ..."
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Cited by 20 (4 self)
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Let p be a fixed prime number, and N be a large integer. The ’Inverse Conjecture for the Gowers norm ’ states that if the ”dth Gowers norm ” of a function f: F N p → F is nonnegligible, that is larger than a constant independent of N, then f can be nontrivially approximated by a degree d − 1 polynomial. The conjecture is known to hold for d = 2, 3 and for any prime p. In this paper we show the conjecture to be false for p = 2 and for d = 4, by presenting an explicit function whose 4th Gowers norm is nonnegligible, but whose correlation any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao [5]. Their analysis uses a modification of a Ramseytype argument of Alon and Beigel [1] to show inapproximability of certain functions by lowdegree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime p, for d = p 2.
Worst case to Average Case Reductions for Polynomials
 the Proceedings of the 49th Annual Symposium on Foundations of Computer Science (FOCS
, 2008
"... A degreed polynomial p in n variables over a field F is equidistributed if it takes on each of its F  values close to equally often, and biased otherwise. We say that p has a low rank if it can be expressed as a bounded combination of polynomials of lower degree. Green and Tao [GT07] have shown t ..."
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Cited by 20 (9 self)
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A degreed polynomial p in n variables over a field F is equidistributed if it takes on each of its F  values close to equally often, and biased otherwise. We say that p has a low rank if it can be expressed as a bounded combination of polynomials of lower degree. Green and Tao [GT07] have shown that bias imply low rank over large fields (i.e. for the case d < F). They have also conjectured that bias imply low rank over general fields. In this work we affirmatively answer their conjecture. Using this result we obtain a general worst case to average case reductions for polynomials. That is, we show that a polynomial that can be approximated by few polynomials of bounded degree, can be computed by few polynomials of bounded degree. We derive some relations between our results to the construction of pseudorandom generators, and to the question of testing concise representations.
Optimal testing of ReedMuller codes
, 2009
"... We consider the problem of testing if a given function ..."
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Cited by 20 (9 self)
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We consider the problem of testing if a given function
On the Power of SmallDepth Computation
, 2009
"... In this work we discuss selected topics on smalldepth computation, presenting a few unpublished proofs along the way. The four chapters contain: 1. A unified treatment of the challenge of exhibiting explicit functions that have small correlation with lowdegree polynomials over {0, 1}. 2. An unpubl ..."
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Cited by 10 (5 self)
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In this work we discuss selected topics on smalldepth computation, presenting a few unpublished proofs along the way. The four chapters contain: 1. A unified treatment of the challenge of exhibiting explicit functions that have small correlation with lowdegree polynomials over {0, 1}. 2. An unpublished proof that small boundeddepth circuits (AC 0) have exponentially small correlation with the parity function. The proof is due to Klivans and Vadhan; it builds upon and simplifies previous ones. 3. Valiant’s simulation of logdepth linearsize circuits of fanin 2 by subexponential size circuits of depth 3 and unbounded fanin. To our knowledge, a proof of this result has never appeared in full.
Testing low complexity affineinvariant properties
, 2013
"... Abstract Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affineinvariant property of multivariate functions over finite fields is testable with a ..."
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Abstract Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affineinvariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the ReedMuller code over F p of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials, except that low degree is preserved by composition with affine maps. The complexity of an affineinvariant property P refers to the maximum complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of linear forms used to characterize P. A more precise statement of our main result is that for any fixed prime p ≥ 2 and fixed integer R ≥ 2, any affineinvariant property P of functions f : F n p → [R] is testable, assuming the complexity of the property is less than p. Our proof involves developing analogs of graphtheoretic techniques in an algebraic setting, using tools from higherorder Fourier analysis.
A fourieranalytic approach to reedmuller decoding
 In Proceedings of the 51th Annual IEEE Symposium on Foundations of Computer Science (FOCS
, 2010
"... Abstract. We present a Fourieranalytic approach to listdecoding ReedMuller codes over arbitrary finite fields. We use this to show that quadratic forms over any field are locally listdecodeable up to their minimum distance. The analogous statement for linear polynomials was proved in the celebr ..."
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Abstract. We present a Fourieranalytic approach to listdecoding ReedMuller codes over arbitrary finite fields. We use this to show that quadratic forms over any field are locally listdecodeable up to their minimum distance. The analogous statement for linear polynomials was proved in the celebrated works of GoldreichLevin [GL89] and GoldreichRubinfeldSudan [GRS00]. Previously, tight bounds for quadratic polynomials were known only for q = 2 and 3 [GKZ08]; the best bound known for other fields was the Johnson radius. Departing from previous work on ReedMuller decoding which relies on some form of selfcorrector [GRS00, AS03, STV01, GKZ08], our work applies ideas from Fourier analysis of Boolean functions to lowdegree polynomials over finite fields, in conjunction with results about the weightdistribution. We believe that the techniques used here could find other applications, we present some applications to testing and learning.