Coin-tossing measures and their Fourier transforms

A coin-tossing measure μ on [0, 1] is a probability measure satisfying μ = ∞ ∗ n=1 pnδ(0) + ( 1− pn)δ(1/2n) where pn ∈ [0, 1], δ(x) denotes the probability atom at x and the convergence is in the weak* sense. We study the asymptotic behavior of averages of the Fourier transform of μ, μˆ(x). For p 2 and ε >0 we prove that |x| R μˆ (x) p dx = O(R1−βp+ε), R→+∞, where βp = 1 −lim sup N→∞ 1 N N n=1 log2 1 + |an|p , an = 2pn − 1. This extends some results due to R. Strichartz for measures which are not self-similar.We also study the Sobolev exponent of | ˆ μ(x)|p and its scaling exponent, as well as the asymptotic behavior of sums of the Walsh–Fourier coefficients of μ.  2004 Elsevier Inc. All rights reserved.