Correlation Testing for Affine Invariant Properties on $\mathbb{F}_p^n$ in the High Error Regime

TL;DR

This paper characterizes affine invariant properties on finite fields that can be tested via correlation, showing they are essentially equivalent to correlation with low-degree polynomials, using higher-order Fourier analysis.

Contribution

It provides a complete classification of correlation testable affine invariant properties over finite fields, extending previous results to non-linear properties.

Findings

Any affine invariant, correlation testable property can be tested by Gowers uniformity norms.

Correlation with the property is equivalent to correlation with degree d polynomials.

The classification applies to both linear and non-linear affine invariant properties.

Abstract

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 $f:\mathbb{F}_p^n \rightarrow \mathbb{F}_p$ with polynomials of degree at most $d \le p$ 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 $d$ 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. 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.