Abstract

We study sequences of functions of the form $\mathbb{F}_p^n \to \{0,1\}$ 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 also 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 Balázs Szegedy [Gowers norms, regularization and limits of functions on abelian groups. 2010. arXiv:1010.6211].

Highlights

  • In limit theories of discrete structures, one often studies a large object by studying its “local statistics”

  • We 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

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Summary

Introduction

In limit theories of discrete structures, one often studies a large object by studying its “local statistics”. Given a subset of Fn, the local information we would like to work with is the distribution of small linear structures, e.g. arithmetic progressions, contained within the set. To this end, we use the sampling rule that considers the restriction of a given function to a random affine subspace. Given a function f : Fn → {0, 1} and a positive integer k, we select a random affine transformation A : Fk → Fn uniformly, and consider the random variable Af : Fk → {0, 1} defined as Af : x → f (Ax) This induces a probability distribution on the set of functions {Fk → {0, 1}}. Before we can state our results we need to recall some results from higher-order Fourier analysis

Basic background
Uniformity norms and non-classical polynomials
Complexity of systems of linear forms
Main results
Proof of the Main Theorem
Necessary depths
Full Text
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