Limits of Boolean Functions on F_pⁿ

2013/08/19 by Hamed Hatami, Pooya Hatami, Hatami, Hamed +3 · 1 citation
Mathematics · #Mathematical Analysis and Transform Methods #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #math.CO #math.FA #msc:46

paper · pdf · doi:10.48550/arxiv.1308.4108

12 pages

arxiv created 2013/08/19 · arxiv updated 2013/08/20

Abstract

We study sequences of functions of the form F_pⁿ -> 0,1 for varying n, and define a notion of convergence based on the induced distributions from restricting the functions to a random affine subspace. Using a decomposition theorem and a recently proven equi-distribution theorem from higher order Fourier analysis, we prove that the limits of such convergent sequences can be represented by certain measurable functions. We are also able to show that every such limit object arises as the limit of some sequence of functions. These results are in the spirit of similar results which have been developed for limits of graph sequences. A more general, albeit substantially more sophisticated, limit object was recently constructed by Szegedy in [Sze10].

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