Testing linear-invariant properties
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We prove that all linear-invariant, linear-subspace hereditary, locally characterized properties are proximity-oblivious testable (with one-sided error and constant query-complexity). In other words, we show that we can distinguish functions F_p^n→[R] satisfying a given property from those that are ϵ-far from satisfying the property as long as the property is definable by restrictions to bounded dimension subspaces. This result can be equivalently stated as an induced arithmetic removal lemma: given a set of colored patterns F_p^d → [R], for every ϵ > 0 there exists δ > 0 such that if a function fF_p^n→[R] has density at most δ of each of the prescribed patterns, then f can be made free of these patterns by recoloring at most ϵ p^n points. The proof of this result uses two main techniques. The first builds upon a long line of work which applies regularity methods, including results from higher order Fourier analysis, to prove results on property testing and removal lemmas. The second, the main innovation of this work, is a novel recoloring technique that allows us to handle an important obstacle encountered by previous works in the arithmetic setting. Roughly speaking, this obstacle is the inability of regularity methods to regularize functions in a neighborhood of the origin.
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