| Summary: | Higher-order Fourier analysis is a powerful tool in the study of problems in additive and extremal combinatorics, for instance the study of arithmetic progressions in primes, where the traditional Fourier analysis comes short. In recent years, higher-order Fourier analysis has found multiple applications in computer science in fields such as property testing and coding theory. In this thesis, we develop new tools within this theory with several new applications such as a characterization theorem in algebraic property testing. One of our main contributions is a strong near-equidistribution result for regular collections of polynomials.
The densities of small linear structures in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by approximating the indicator function of a subset by a function of bounded number of polynomials. Then, to approximate the average, it suffices to know the joint distribution of the polynomials applied to the linear forms. We prove a near-equidistribution theorem that describes these distributions for the group F(n/p) when p is a fixed prime. This fundamental fact was previously known only under various extra assumptions about the linear forms or the field size. We use this near-equidistribution theorem to settle a conjecture of Gowers and Wolf on the true complexity of systems of linear forms.
Our next application is towards a characterization of testable algebraic properties. We prove that every locally characterized affine-invariant property of functions f : F(n/p) ā R with nā N, is testable. In fact, we prove that any such property P is proximity-obliviously testable. More generally, we show that any affine-invariant property that is closed under subspace restrictions 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.
We discuss several notions of regularity which allow us to deduce algorithmic versions of various regularity lemmas for polynomials by Green and Tao and by Kaufman and Lovett. We show that our algorithmic regularity lemmas for polynomials imply algorithmic versions of several results relying on regularity, such as decoding Reed-Muller codes beyond the list decoding radius (for certain structured errors), and prescribed polynomial decompositions.
Finally, motivated by the definition of Gowers norms, we investigate norms defined by different systems of linear forms. We give necessary conditions on the structure of systems of linear forms that define norms. We prove that such norms can be one of only two types, and assuming that |F p| is sufficiently large, they essentially are equivalent to either a Gowers norm or Lp norms.
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