FKN Theorem on the biased cube

Piotr Nayar

Colloquium Mathematicae, Tome 135 (2014), p. 253-261 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2014-01-01

Zbl 1309.42040

EUDML-ID : urn:eudml:doc:284353