Gowers Norm, Function Limits, and Parameter Estimation

TL;DR

This paper introduces a new metric based on the Gowers norm to analyze limits of function sequences over finite fields, characterizes affine-invariant parameters, and demonstrates constant-query testability of certain polynomial properties.

Contribution

It defines a metric on function limits inspired by the Gowers norm and characterizes constant-query estimable affine-invariant parameters, enabling new property testing results.

Findings

Introduces a metric over function limits using Gowers norm

Characterizes affine-invariant parameters that are constant-query estimable

Shows certain polynomial properties are constant-query testable

Abstract

Let $\{f_i:\mathbb{F}_p^i \to \{0,1\}\}$ be a sequence of functions, where $p$ is a fixed prime and $\mathbb{F}_p$ is the finite field of order $p$. The limit of the sequence can be syntactically defined using the notion of ultralimit. Inspired by the Gowers norm, we introduce a metric over limits of function sequences, and study properties of it. One application of this metric is that it provides a characterization of affine-invariant parameters of functions that are constant-query estimable. Using this characterization, we show that the property of being a function of a constant number of low-degree polynomials and a constant number of factored polynomials (of arbitrary degrees) is constant-query testable if it is closed under blowing-up. Examples of this property include the property of having a constant spectral norm and degree-structural properties with rank conditions.