FKN theorem on the biased cube

Piotr Nayar

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

We consider Boolean functions defined on the discrete cube {-γ, γ-1}n equipped with a product probability measure µ⊗n, where µ = βδ-γ + αδγ-1 and γ = α/β. This normalization ensures that the coordinate functions (xi)i=1,…,n are orthonor-mal in L2({-γ, γ-1}n, µ⊗n). We prove that if the spectrum of a Boolean function is con-centrated on the first two Fourier levels, then the function is close to a certain function of one variable. Our theorem strengthens the non-symmetric FKN Theorem due to Jendrej, Oleszkiewicz and Wojtaszczyk. Moreover, in the symmetric case α = β = 1/2 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.

Original languageEnglish (US)
Pages (from-to)253-261
Number of pages9
JournalColloquium Mathematicum
Volume137
Issue number2
DOIs
StatePublished - Jan 1 2014

Keywords

  • Boolean functions
  • FKN theorem
  • Walsh-fourier expansion

Fingerprint Dive into the research topics of 'FKN theorem on the biased cube'. Together they form a unique fingerprint.

  • Cite this