FKN Theorem on the biased cube

• Autorzy: Piotr Nayar
• Języki publikacji: EN

Abstrakty

EN

We consider Boolean functions defined on the discrete cube ${-γ, γ^{-1}}ⁿ$ equipped with a product probability measure $μ^{⊗ n}$, where $μ = βδ_{-γ} + αδ_{γ^{-1}}$ and γ = √(α/β). This normalization ensures that the coordinate functions $(x_i)_{i=1,...,n}$ are orthonormal in $L₂({-γ,γ^{-1}}ⁿ,μ^{⊗ n})$. We prove that if the spectrum of a Boolean function is concentrated 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.

Kategorie tematyczne

• 60E15: Inequalities; stochastic orderings
• 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2014
• Tom: 137
• Numer: 2
• Strony: 253-261

Daty

wydano

2014

Twórcy

autor

Piotr Nayar

• Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
• Institute for Mathematics and its Applications, College of Science and Engineering, University of Minnesota, 207 Church Street SE, 306 Lind Hall, Minneapolis, MN 55455, U.S.A.

Identyfikatory

DOI

10.4064/cm137-2-9