On the Fourier-walsh Spectrum of the Moebius Function

We study the Fourier-Walsh spectrum {μ̂(S);S ⊂ {1, . . . , n}} of the Moebius function μ restricted to {0, 1, 2, . . . , 2 − 1} ≃ {0, 1} and prove that it is not captured by levels {μ̂(S)| |S| < n 2 3 −ε}. An application to correlation with monotone Bolean functions is given. 0. Introduction This paper may be seen as a companion of [B1] on the behavior of the Fourier-Walsh coefficients of the Moebius function μ restricted to a large interval {1, 2, . . . , N}, N = 2. While in [B1] we did establish nontrivial uniform upper bounds on the F-W coefficients of μ, we are interested here in their distribution. Our main result shows that μ cannot be captured by ‘low order’ Walsh functions, more precisely Theorem 1. Let λ > 0 be a fixed constant and n0 ∼ (logn)n. Then ∑ |A|≤n0 |μ̂(A)| < (logn) (0.1) where μ̂(A) = 1 N ∑ x∈{0,1}n wA(x)μ ( n−1 ∑