# Finitary Boolean functions

## TL;DR

This paper extends the analysis of Boolean functions to infinite-dimensional settings, introducing noise sensitivity, Fourier analysis, and influence bounds for finitary functions, with applications to models like the voter model on sparse graphs.

## Contribution

It generalizes key concepts like noise sensitivity and influence bounds to finitary Boolean functions on infinite-dimensional spaces, providing new theorems and applications.

## Key findings

- Extended noise sensitivity to infinite-dimensional Boolean functions.
- Established hypercontractivity and influence bounds in this setting.
- Demonstrated noise sensitivity of the voter model on sparse graphs.

## Abstract

We study functions on the infinite-dimensional Hamming cube $\{-1,1\}^\infty$, in particular Boolean functions into $\{-1,1\}$, generalising results on analysis of Boolean functions on $\{-1,1\}^n$ for $n\in\mathbb{N}$. The notion of noise sensitivity, first studied in arXiv:math/9811157 , is extended to this setting, and basic Fourier formulas are established. We also prove hypercontractivity estimates for these functions, and give a version of the Kahn-Kalai-Linial theorem giving a bound relating the total influence to the maximal influence. Particular attention is paid to so-called finitary functions, which are functions for which there exists an algorithm that almost surely queries only finitely many bits. Two versions of the Benjamini-Kalai-Schramm theorem characterizing noise sensitivity in terms of the sum of squared influences are given, under additional moment hypotheses on the amount of bits looked at by an algorithm. A version of the Kahn-Kalai-Linial theorem giving that the maximal influence is of order $\frac{\log(n)}{n}$ is also given, replacing $n$ with the expected number of bits looked at by an algorithm. Finally, we show that the result in arXiv:math/0504586 that revealments going to zero implies noise sensitivity also holds for finitary functions, and apply this to show noise sensitivity of a version of the voter model on sufficiently sparse graphs.