Limits of Boolean Functions on F_p^n

TL;DR

This paper investigates the limits of Boolean functions on finite fields, establishing a framework for their convergence and representation using advanced Fourier analysis, akin to graph limit theories.

Contribution

It introduces a new notion of convergence for Boolean functions on F_p^n and characterizes their limit objects using higher order Fourier analysis techniques.

Findings

Limit objects can be represented by measurable functions.

Every limit object arises from a sequence of Boolean functions.

The results extend the theory of graph limits to Boolean functions.

Abstract

We study sequences of functions of the form F_p^n -> {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].