Testing Linear-Invariant Properties

Fix a prime $p$ and positive integer $R$. We study the property testing of functions $\mathbb F_p^n\to[R]$. say that is testable if there exists an oblivious tester for this with one-sided error constant query complexity. Furthermore, proximity oblivious-testable (PO-testable) test also independent parameter $\epsilon$. It known number natural properties such as linearity being low degree polynomial are PO-testable. These examples linear-invariant properties, meaning they preserved under linear automorphisms domain. Following work Kaufman Sudan, has been important problem in arithmetic testing. A central conjecture field, proposed by Bhattacharyya, Grigorescu, Shapira, only it semi subspace-hereditary. prove two results, first resolves second classifies PO-testable properties. (1) (2) locally characterized. Our innovations two-fold. give more powerful version compactness argument introduced Alon Shapira. This relies on new strong regularity lemma which one mixes different levels Gowers uniformity. allows us to extend Fischer, Hatami, Lovett removing bounded complexity restriction their work. innovation novel recoloring technique called patching. Ramsey-theoretic critical working setting remove translation-invariant present previous