Gowers Norm, Function Limits, and Parameter Estimation

Let $\{f_i:\mathbb{F}_p^i \to \{0,1\}\}$ be a sequence of functions, where $p$ is a fixed prime and $\mathbb{F}_p$ is the finite field of order $p$. The limit of the sequence can be syntactically defined using the notion of ultralimit. Inspired by the Gowers norm, we introduce a metric over limits of function sequences, and study properties of it. One application of this metric is that it provides a characterization of affine-invariant parameters of functions that are constant-query estimable. Using this characterization, we show that the property of being a function of a constant number of low-degree polynomials and a constant number of factored polynomials (of arbitrary degrees) is constant-query testable if it is closed under blowing-up. Examples of this property include the property of having a constant spectral norm and degree-structural properties with rank conditions.Comment: arXiv admin note: text overlap with arXiv:1212.3849, arXiv:1308.4108 by other author

Testing Low Complexity Affine-Invariant Properties

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 affine-invariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the Reed-Muller 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 affine-invariant 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 affine-invariant property P of functions f: F_p^n -> [R] is testable, assuming the complexity of the property is less than p. Our proof involves developing analogs of graph-theoretic techniques in an algebraic setting, using tools from higher-order Fourier analysis.Comment: 38 pages, appears in SODA '1

Every locally characterized affine-invariant property is testable

Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-invariant property having local characterizations are testable. In fact, we show a proximity-oblivious test for any such property P, meaning that there is a test that, given an input function f, makes a constant number of queries to f, always accepts if f satisfies P, and rejects with positive probability if the distance between f and P is nonzero. 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 low-degree 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 low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties

(Uniform) convergence of twisted ergodic averages

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Thue--Morse along the sequence of cubes

The Thue--Morse sequence $t=01101001\cdots$ is an automatic sequence over the alphabet $\{0,1\}$. It can be defined as the binary sum-of-digits function $s:\mathbb N\rightarrow\mathbb N$, reduced modulo $2$, or by using the substitution $0\mapsto 01$, $1\mapsto 10$. We prove that the asymptotic density of the set of natural numbers $n$ satisfying $t(n^3)=0$ equals $1/2$. Comparable results, featuring asymptotic equivalence along a polynomial as in our theorem, were previously only known for the linear case [A. O. Gelfond, Acta Arith. 13 (1967/68), 259--265], and for the sequence of squares. The main theorem in [C. Mauduit and J. Rivat, Acta Math. 203 (2009), no. 1, 107--148] was the first such result for the sequence of squares. Concerning the sum-of-digits function along polynomials $p$ of degree at least three, previous results were restricted either to lower bounds (such as for the numbers $\#\{n<N:t(p(n))=0\}$), or to sum-of-digits functions in ``sufficiently large bases''. By proving an asymptotic equivalence for the case of the Thue--Morse sequence, and a cubic polynomial, we move one step closer to the solution of the third Gelfond problem on the sum-of-digits function (1967/1968), op. cit.Comment: 50 pages. Corrected several small inconsistencies present in the first version; reworked the articl