Self-testing/correcting with applications to numerical problems
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
Robust Characterizations of Polynomials and Their Applications to Program Testing
Robust Characterizations of Polynomials and Their Applications to Program Testing
Algebraic property testing: the role of invariance
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Worst Case to Average Case Reductions for Polynomials
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
On proximity oblivious testing
Proceedings of the forty-first annual ACM symposium on Theory of computing
Every locally characterized affine-invariant property is testable
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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Recently there has been much interest in Gowers uniformity norms from the perspective of theoretical computer science. This is mainly due to the fact that these norms provide a method for testing whether the maximum correlation of a function f:Fpn - Fp with polynomials of degree at most d ≤ p is non-negligible, while making only a constant number of queries to the function. This is an instance of correlation testing. In this framework, a fixed test is applied to a function, and the acceptance probability of the test is dependent on the correlation of the function from the property. This is an analog of proximity oblivious testing, a notion coined by Goldreich and Ron, in the high error regime. We study in this work general properties which are affine invariant and which are correlation testable using a constant number of queries. We show that any such property (as long as the field size is not too small) can in fact be tested by the Gowers uniformity test, and hence having correlation with the property is equivalent to having correlation with degree d polynomials for some fixed d. We stress that our result holds also for non-linear properties which are affine invariant. This completely classifies affine invariant properties which are correlation testable. The proof is based on higher-order Fourier analysis, where we establish a new approximate orthogonality for structures defined by linear forms. In particular, this resolves an open problem posed by Gowers and Wolf. Another ingredient is a nontrivial extension of a graph theoretical theorem of Erdos, Lovasz and Spencer to the context of additive number theory.