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Correlation Testing for Affine Invariant Properties on in the High Error Regime
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 with polynomials of
degree at most is non-negligible, while making only a constant number
of queries to the function. This is an instance of {\em 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 {\em proximity oblivious testing}, a notion
coined by Goldreich and Ron, in the high error regime. In this work, we study
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 Gowers
uniformity tests, and hence having correlation with the property is equivalent
to having correlation with degree polynomials for some fixed . 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.
Another ingredient is a nontrivial extension of a graph theoretical theorem of
Erd\"os, Lov\'asz and Spencer to the context of additive number theory.Comment: 43 pages. A preliminary version of this work appeared in STOC' 201