FKN Theorem on the biased cube

TL;DR

This paper extends the FKN theorem to biased discrete cubes, showing that functions with low Fourier spectrum concentration are close to functions of one variable, with additional results for symmetric cases and affine functions.

Contribution

It generalizes the FKN theorem to biased measures and symmetric functions, providing new stability results for Boolean functions on the discrete cube.

Findings

Functions with spectrum on first two Fourier levels are close to single-variable functions.

In symmetric cases, functions close to affine functions are also close to bounded affine functions.

The results apply to functions defined on biased product measures.

Abstract

In this note we consider Boolean functions defined on the discrete cube equipped with a biased product probability measure. We prove that if the spectrum of such a function is concentrated on the first two Fourier levels, then the function is close to a certain function of one variable. Moreover, in the symmetric case we prove that if a [-1,1]-valued function defined on the discrete cube is close to a certain affine function, then it is also close to a [-1,1]-valued affine function.