On the Fourier-Walsh spectrum of the Moebius function

Abstract

We study the Fourier-Walsh spectrum $$\{ \widehat \mu (S);S \subset \{ 1, \ldots ,n\} \} $$ of the Moebius function µ restricted to $$\{ 0,1,2, \ldots ,{2^n} - 1\} \simeq {\{ 0,1\} ^n}$$ and prove that it is not captured by levels $$\{ \widehat \mu (S)||S| < {n^{\frac{2}{3} - \varepsilon }}\} $$ . An application to correlation with monotone Boolean functions is given.