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].
@article{10_37236_4445,
author = {Hamed Hatami and Pooya Hatami and James Hirst},
title = {Limits of {Boolean} functions on {\(\mathbb{F}_p^n\)}},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {4},
doi = {10.37236/4445},
zbl = {1364.11152},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4445/}
}
TY - JOUR
AU - Hamed Hatami
AU - Pooya Hatami
AU - James Hirst
TI - Limits of Boolean functions on \(\mathbb{F}_p^n\)
JO - The electronic journal of combinatorics
PY - 2014
VL - 21
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/4445/
DO - 10.37236/4445
ID - 10_37236_4445
ER -
%0 Journal Article
%A Hamed Hatami
%A Pooya Hatami
%A James Hirst
%T Limits of Boolean functions on \(\mathbb{F}_p^n\)
%J The electronic journal of combinatorics
%D 2014
%V 21
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/4445/
%R 10.37236/4445
%F 10_37236_4445
Hamed Hatami; Pooya Hatami; James Hirst. Limits of Boolean functions on \(\mathbb{F}_p^n\). The electronic journal of combinatorics, Tome 21 (2014) no. 4. doi: 10.37236/4445
