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39
Optimal testing of ReedMuller codes
, 2009
"... We consider the problem of testing if a given function ..."
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Cited by 20 (9 self)
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We consider the problem of testing if a given function
AN EQUIVALENCE BETWEEN INVERSE SUMSET THEOREMS AND INVERSE CONJECTURES FOR THE U³ NORM
, 2009
"... We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theore ..."
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We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theorems of Freĭman type are equivalent to the known inverse results for the Gowers U 3 norms, and moreover that the conjectured polynomial strengthening of the former is also equivalent to the polynomial strengthening of the latter. We establish this equivalence in two model settings, namely that of the finite field vector spaces Fn 2, and of the cyclic groups Z/NZ. In both cases the argument involves clarifying the structure of certain types of approximate homomorphism.
Testing low complexity affineinvariant properties
, 2013
"... Abstract Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affineinvariant property of multivariate functions over finite fields is testable with a ..."
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Abstract Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affineinvariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the ReedMuller code over F p of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials, except that low degree is preserved by composition with affine maps. The complexity of an affineinvariant property P refers to the maximum complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of linear forms used to characterize P. A more precise statement of our main result is that for any fixed prime p ≥ 2 and fixed integer R ≥ 2, any affineinvariant property P of functions f : F n p → [R] is testable, assuming the complexity of the property is less than p. Our proof involves developing analogs of graphtheoretic techniques in an algebraic setting, using tools from higherorder Fourier analysis.
Every locally characterized affineinvariant property is testable
, 2013
"... Let F = Fp for any fixed prime p> 2. An affineinvariant property is a property of functions on Fn that is closed under taking affine transformations of the domain. We prove that all affineinvariant properties that have local characterizations are testable. In fact, we give a proximityoblivious ..."
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Let F = Fp for any fixed prime p> 2. An affineinvariant property is a property of functions on Fn that is closed under taking affine transformations of the domain. We prove that all affineinvariant properties that have local characterizations are testable. In fact, we give a proximityoblivious 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 affineinvariant 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 being decomposable into a known structure of lowdegree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degreed polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degreestructural properties and are therefore locally characterized. Our results use 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 lowdegree nonclassical polynomials. We establish a new equidistribution result for high rank nonclassical polynomials that drives the proofs of both the testability results and the local characterization of degreestructural properties.