Self-testing/correcting with applications to numerical problems
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
Property testing and its connection to learning and approximation
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Lower Bounds for Approximations by Low Degree Polynomials Over Z_m
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Testing Polynomials over General Fields
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Low-degree tests at large distances
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Norms, XOR Lemmas, and Lower Bounds for GF(2) Polynomials and Multiparty Protocols
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Structure and Randomness in Combinatorics
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Pseudorandom Bits for Polynomials
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Linearity testing in characteristic two
IEEE Transactions on Information Theory - Part 1
Unconditional pseudorandom generators for low degree polynomials
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Small Sample Spaces Cannot Fool Low Degree Polynomials
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Guest Column: correlation bounds for polynomials over {0 1}
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Foundations and Trends® in Theoretical Computer Science
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SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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SIAM Journal on Computing
Optimal Testing of Reed-Muller Code
Property testing
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Property testing
Every locally characterized affine-invariant property is testable
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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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 "d-th Gowers norm" of a function f:FNp to Fp is non-negligible, that is larger than a constant independent of N, then f has a non-trivial correlation with 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 4-th Gowers norm is non-negligible, but whose correlation with any polynomial of degree 3 is exponentially small. Essentially the same result, with different bounds for correlation, was independently obtained by Green and Tao. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel to show inapproximability of certain functions by low-degree 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 = p2.