Every locally characterized affine-invariant property is testable

Abstract

Set F = Fp for any fixed prime p2. An affine-invariant property is a property of functions over F n that is closed under taking affine transformations of the domain. We prove that all affine-invariant properties having local characterizations are testable. In fact, we show a proximity-oblivious test for any such property P, meaning that given an input function f , we make a constant number of queries to f , always accept if f satisfies P, and otherwise reject with probability larger than a positive number that depends only on the distance between f and P. More generally, we show that any affine-invariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable.We also prove that any property that can be described as the property of decomposing into a known structure of lowdegree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized.Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of a small number of low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the *

Stability and exact Turán numbers for matroids

“…The first result we state is the Strong Decomposition Theorem of [1], a regularity result on binary matroids analogous to the Szemerédi regularity lemma for graphs. Stating the result precisely requires the introduction of several technical definitions not useful in the rest of the paper.…”

“…This paper deals with Turán-type problems in the arithmetic setting of subsets of F n 2 , where we fix a set N ⊆ F k 2 and consider, for n much larger than k, the size of a set M ⊆ F n 2 that does not contain any subset that is the image of N under an injective linear map ϕ : F k 2 → F n 2 . This is analogous to excluding a fixed subgraph H from a graph G. Such problems have been considered both in the language of arithmetic combinatorics [1,9,10] and equivalently matroid theory [6,7,15]; here we will use the term 'matroid' for brevity to describe the relevant notions of containment and isomorphism, as well as to highlight the strong analogies with graph theory, and to describe the 'host' object M and the 'system of linear forms' being excluded in a unified way. While they are stated combinatorially, all of our results depend on a Fourieranalytic regularity lemma of Hatami et al [10] and a new associated counting lemma due to the second author [11].…”

Testing Linear-Invariant Function Isomorphism

“…Very recently, [21] gave a characterization of affine-invariant properties that are constant-query testable with one-sided error. Their work does not derive our result since having small spectral norm and satisfying the condition they gave are incomparable.…”

Tight Lower Bounds for Testing Linear Isomorphism

“…Alon et al [2] and Borgs et al [15] already showed a complete characterization of strongly testable properties of dense graphs. Very recently, Bhattacharyya et al [8] announced a characterization of one-sided strongly testable boolean families that are invariant under "affine" transformations of the domain.…”

“…Similarly, P is affine-invariant if f ∈ P if and only if the function f • L C,b ∈ P, for any affine L C,b , where f • L C,b (x) = f (L C,b (x)). Following [30] linear/affine invariant families have been intensely studied on two fronts: properties that arise in the setting of linear codes [30,28,27,6,4,3,31,5,29], and properties that arise more often in the study of boolean functions [26,32,40,9,8]. All these works study properties that are testable with a constant number of queries.…”