FKN theorem on the biased cube

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.