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- W1543632961 abstract "AbstractWe study sequences of functions of the form F np → {0,1} for varying n, and define a notionof convergence based on the induced distributions from restricting the functions to a randomaffine subspace. Using a decomposition theorem and a recently proven equi-distribution theoremfrom higher order Fourier analysis, we prove that the limits of such convergent sequences canbe represented by certain measurable functions. We are also able to show that every suchlimit object arises as the limit of some sequence of functions. These results are in the spirit ofsimilar results which have been developed for limits of graph sequences. A more general, albeitsubstantially more sophisticated, limit object was recently constructed by Szegedy in [Sze10]. 1 Introduction In limit theories of discrete structures, one often studies a large object by studying its “localstatistics”. More precisely, there is a sampling rulethat allows one to sample a randomsubstructure,and this induces a probability measure on the set of possible small substructures. For example givena graph G and a positive integer k, one can select k random vertices in G and look at the subgraphinduced by G on these k vertices. This introduces a probability distribution on k-vertex graphs.Every such sampling rule leads to a notion of convergence. Namely a sequence of structures iscalled convergent if these probability distributions converge. So in the above example, a sequenceof graphs is called convergent [LS06] if for every k, the corresponding probability distributions onthe k-vertex graphs converges.Let p be a fixed prime, and denote F= F" @default.
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- W1543632961 date "2014-01-01" @default.
- W1543632961 modified "2023-09-27" @default.
- W1543632961 title "Limits of Boolean Functions on F n" @default.
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