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Higher order Fourier analysis as an algebraic theory I

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arxiv 0903.0897 v1 pith:T3K4W7A4 submitted 2009-03-05 math.CO math.DS

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keywords theoryabelianalgebraicgroupshigherorderanalysisapproaches
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Ergodic theory, Higher order Fourier analysis and the hyper graph regularity method are three possible approaches to Szemer\'edi type theorems in abelian groups. In this paper we develop an algebraic theory that creates a connection between these approaches. Our main method is to take the ultra product of abelian groups and to develop a precise algebraic theory of higher order characters on it. These results then can be turned back into approximative statements about finite Abelian groups.

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  1. Spectral algorithms in higher-order Fourier analysis

    math.CO 2025-01 conditional novelty 8.0 of 10

    A spectral inverse theorem and a spectral regularity theorem show that leading eigenvectors of Fourier-denoised matrices recover quadratic Fourier structure, giving new algorithms for quadratic denoising and character...

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