Boolean functions whose Fourier transform is concentrated on pairwise disjoint subsets of the input

TL;DR

This paper proves that Boolean functions close to sums of independent functions or variables are essentially close to a single such component, generalizing the Friedgut-Kalai-Naor theorem and showing the tightness of variance dependence.

Contribution

It extends the FKN theorem to functions with Fourier transforms concentrated on disjoint subsets, establishing variance-dependent bounds for Boolean functions.

Findings

Functions close to sums of independent components are near a single component.

Results hold regardless of the number of variables, depending only on variance.

Variance dependence in the approximation is proven to be tight.

Abstract

We consider Boolean functions f:{-1,1}^n->{-1,1} that are close to a sum of independent functions on mutually exclusive subsets of the variables. We prove that any such function is close to just a single function on a single subset. We also consider Boolean functions f:R^n->{-1,1} that are close, with respect to any product distribution over R^n, to a sum of their variables. We prove that any such function is close to one of the variables. Both our results are independent of the number of variables, but depend on the variance of f. I.e., if f is \epsilon*Var(f)-close to a sum of independent functions or random variables, then it is O(\epsilon)-close to one of the independent functions or random variables, respectively. We prove that this dependence on Var(f) is tight. Our results are a generalization of the Friedgut-Kalai-Naor Theorem [FKN'02], which holds for functions f:{-1,1}^n->{-1,1} that are close to a linear combination of uniformly distributed Boolean variables.