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The distribution of polynomials over finite fields, with applications to the Gowers norms
, 2007
"... In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P: F n → F is poorlydistributed only if P is determined by the values of a few polynomials of lower ..."
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Cited by 40 (2 self)
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In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P: F n → F is poorlydistributed only if P is determined by the values of a few polynomials of lower degree, in which case we say that P has small rank. We give several applications of this result, paying particular attention to consequences for the theory of the socalled Gowers norms. We establish an inverse result for the Gowers U d+1norm of functions of the form f(x) = eF(P(x)), where P: F n → F is a polynomial of degree less than F, showing that this norm can only be large if f correlates with eF(Q(x)) for some polynomial Q: F n → F of degree at most d. The requirement deg(P) < F  cannot be dropped entirely. Indeed, we show the above claim fails in characteristic 2 when d = 3 and deg(P) = 4, showing that the quartic symmetric polynomial S4 in F n 2 has large Gowers U 4norm but does not correlate strongly with any cubic polynomial. This shows that the theory of Gowers norms in low characteristic is not as simple as previously supposed. This counterexample has also been discovered independently by Lovett, Meshulam, and Samorodnitsky [15]. We conclude with sundry other applications of our main result, including a recurrence result and a certain type of nullstellensatz.
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.
A hypergraph dictatorship test with perfect completeness
 In APPROXRANDOM
, 2009
"... A hypergraph dictatorship test is first introduced by Samorodnitsky and Trevisan in [21] and serves as a key component in their unique games based PCP construction. Such a test has oracle access to a collection of functions and determines whether all the functions are the same dictatorship, or all t ..."
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Cited by 1 (1 self)
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A hypergraph dictatorship test is first introduced by Samorodnitsky and Trevisan in [21] and serves as a key component in their unique games based PCP construction. Such a test has oracle access to a collection of functions and determines whether all the functions are the same dictatorship, or all their low degree influences are o(1). The test in [21] makes q ≥ 3 queries and has amortized query complexity log q
The Gowers Norm in the Testing of Boolean Functions
, 2009
"... A property tester is a fast, randomized algorithm that reads only a few entries of the input, and based on the values of these entries, it distinguishes whether the input has a certain property or is “different ” from any input having this property. Furthermore, we say that a property tester has com ..."
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A property tester is a fast, randomized algorithm that reads only a few entries of the input, and based on the values of these entries, it distinguishes whether the input has a certain property or is “different ” from any input having this property. Furthermore, we say that a property tester has completeness c and soundness s if it accepts all inputs having the property with probability at least c and accepts “different ” inputs with probability at most s+ o(1). In this thesis we present two property testers for boolean functions on the boolean cube {0, 1}n. We summarize our contribution as follows. • We present a new dictatorship test that determines whether the function is a dictator (of the form f(x) = xi for some coordinate i), or a function that is an “antidictator. ” Our test is “adaptive, ” makes q queries, has completeness 1, and soundness O(q3) · 2−q. Previously, a dictatorship test that has sound
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"... Additive combinatorics is the branch of combinatorics where the objects of study are subsets of the integers or of other abelian groups, and one is interested in properties and patterns that can be expressed in terms of linear equations. More generally, arithmetic combinatorics deals with properties ..."
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Additive combinatorics is the branch of combinatorics where the objects of study are subsets of the integers or of other abelian groups, and one is interested in properties and patterns that can be expressed in terms of linear equations. More generally, arithmetic combinatorics deals with properties and patterns that can be expressed via additions and multiplications. In the past ten years, additive and arithmetic combinatorics have been extremely successful areas of mathematics, featuring a convergence of techniques from graph theory, analysis and ergodic theory. They have helped prove longstanding open questions in additive number theory, and they offer much promise of future progress. Techniques from additive and arithmetic combinatorics have found several applications in computer science too, to property testing, pseudorandomness, PCP constructions, lower bounds, and extractor constructions. Typically, whenever a technique from additive or arithmetic combinatorics becomes understood by computer scientists, it finds some application. Considering that there is still a lot of additive and arithmetic combinatorics that computer scientists do not understand (and, the field being very active, even more will be developed in the near future), there seems to be much potential for future connections and applications.
www.theoryofcomputing.org Inverse Conjecture for the Gowers Norm is False
"... Abstract: 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 → Fp is nonnegligible, that is, larger than a constant independent of N, then f is nontrivially correlated to a degree(d − ..."
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Abstract: 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 → Fp is nonnegligible, that is, larger than a constant independent of N, then f is nontrivially correlated to 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 d = 4, by presenting an explicit function whose 4th Gowers norm is nonnegligible, but whose correlation to any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao (2009). ACM Classification: F.2.2 AMS Classification: 05E99 Key words and phrases: Inverse Gowers conjecture, additive combinatorics, Gowers norm
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"... Over the last few decades, there have been tremendous advances in computing technology in our daily life. We are witnessing computing devices that are not only increasing in speed and storage, but also becoming more mobile, userfriendly, and widely connected. Spurred by these hardware advances, the ..."
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Over the last few decades, there have been tremendous advances in computing technology in our daily life. We are witnessing computing devices that are not only increasing in speed and storage, but also becoming more mobile, userfriendly, and widely connected. Spurred by these hardware advances, there has been an explosion of data in the last decade, due to increased digitization of information that are aggregated from a highly interconnected network of a growing number of users. The prevalence of these massive datasets requires a refined notion of how we measure the efficiency of an algorithm. Traditionally, algorithms that run in polynomial time are considered practical, and lineartime algorithms are the paradigm of efficiency. However, when working with huge datasets, especially those arising from the Internet, reading the input in its entirety may no longer be feasible. My research is motivated by understanding what algorithms can do when only a sublinear portion of the input is examined. Property testing The field of property testing, initiated by Blum, Luby, and Rubinfeld [3] is concerned with these sublinear algorithms that examine the input at a few select entries, and based on the values of these entries, decide whether the input satisfies a certain property or “looks different ” from any input that satisfies this property. In other words, a testing algorithm decides whether the data possesses the desired property or not.