Bohr's phenomenon for functions on the Boolean cube

TL;DR

This paper investigates the decay of Fourier spectra of functions on the Boolean cube, defining a Boolean radius to measure the spectral decay, and analyzes its asymptotic behavior across various function classes, revealing parallels and differences with complex analysis.

Contribution

It provides the first precise asymptotic analysis of the Boolean radius for multiple natural classes of Boolean functions, extending Bohr's phenomenon to the discrete setting.

Findings

Asymptotic behavior of Boolean radius for all functions on the cube

Boolean radius for homogeneous functions

Boolean radius for threshold functions

Abstract

We study the asymptotic decay of the Fourier spectrum of real functions $f\colon \{-1,1\}^N \rightarrow \mathbb{R}$ in the spirit of Bohr's phenomenon from complex analysis. Every such function admits a canonical representation through its Fourier-Walsh expansion $f(x) = \sum_{S\subset \{1,\ldots,N\}}\widehat{f}(S) x^S \,,$ where $x^S = \prod_{k \in S} x_k$. Given a class $\mathcal{F}$ of functions on the Boolean cube $\{-1, 1\}^{N} $, the Boolean radius of $\mathcal{F}$ is defined to be the largest $\rho \geq 0$ such that $\sum_{S}{|\widehat{f}(S)| \rho^{|S|}} \leq \|f\|_{\infty}$ for every $f \in \mathcal{F}$. We give the precise asymptotic behaviour of the Boolean radius of several natural subclasses of functions on finite Boolean cubes, as e.g. the class of all real functions on $\{-1, 1\}^{N}$, the subclass made of all homogeneous functions or certain threshold functions. Compared with the classical complex situation subtle differences as well as striking parallels occur.