FKN theorem on the biased cube

Piotr Nayar

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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
Issue number2
StatePublished - 2014

Bibliographical note

Publisher Copyright:
© Instytut Matematyczny PAN, 2014.


  • Boolean functions
  • FKN theorem
  • Walsh-fourier expansion


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