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FKN Theorem on the biased cube

Piotr Nayar — 2014

Colloquium Mathematicae

We consider Boolean functions defined on the discrete cube - γ , γ - 1 equipped with a product probability measure μ n , where μ = β δ - γ + α δ γ - 1 and γ = √(α/β). This normalization ensures that the coordinate functions ( x i ) i = 1 , . . . , n are orthonormal in L ( - γ , γ - 1 , 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...

A Note on the Rational Cuspidal Curves

Piotr NayarBarbara Pilat — 2014

Bulletin of the Polish Academy of Sciences. Mathematics

In this short note we give an elementary combinatorial argument, showing that the conjecture of J. Fernández de Bobadilla, I. Luengo-Velasco, A. Melle-Hernández and A. Némethi [Proc. London Math. Soc. 92 (2006), 99-138, Conjecture 1] follows from Theorem 5.4 of Brodzik and Livingston [arXiv:1304.1062] in the case of rational cuspidal curves with two critical points.

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