Limits of Boolean Functions on $\mathbb{F}_p^n$

  • Hamed Hatami
  • Pooya Hatami
  • James Hirst
Keywords: Higher order Fourier analysis, limit theory


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].

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