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

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.06793
 Bibcode:
 2019arXiv191106793T
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Computational Complexity
 EPrint:
 40 pages