Limits of Boolean Functions over Finite Fields

In this thesis, we study sequences of functions of the form Fp → {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. One of the key tools we use is the recently developed theory of ‘higher order Fourier analysis’, where the characters of standard Fourier analysis are replaced with exponentials of higher degree polynomials. This is not a trivial extension by any means, but when the polynomials are chosen with some care, the higher order decomposition can be taken to have properties analogous to those of the classical Fourier transform. The result of applying higher order Fourier analysis in this setting is the necessity to determine the distribution of a collection of polynomials when they are composed with some additional linear structures. Here, we make use of a recently proven equidistribution theorem, relying on a near-orthogonality result showing that the higher order characters can be made orthogonal up to an arbitrarily small error term. With these tools, we prove that the limit of every convergent sequence of functions can be represented by a limit object which takes the form of a certain measurable function on a group we construct. We also show that every such limit object arises as the limit of some sequence of functions. These results are in the spirit of analogous 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].