Asymptotic behavior of Fourier transforms of self-similar measures

Abstract:

Let $\mu$ be a self-similar probability measure on $\mathbb {R}$ satisfying $\mu =\sum _{j=1}^mp_j\mu \circ F_j^{-1},$ where $F_j(x)=\rho x+a_j,$ $0<$ $\rho <1,$ $a_j\in \mathbb {R},$ $p_j>0$ and $\sum _{j=1}^mp_j=1.$ Let $\hat {\mu } (t)$ be the Fourier transform of $\mu .$ A necessary and sufficient condition for $\hat {\mu }(t)$ to approach zero at infinity is given. In particular, if $a_j=j$ and $p_j=1/m$ for $j=1,...,m,$ then $\lim \sup _{t\rightarrow \infty }|\hat {\mu }(t)|>0$ if and only if $\rho ^{-1}$ is a PV-number and $\rho ^{-1}$ is not a factor of $m$. This generalizes the corresponding theorem of Erdös and Salem for the case $m=2.$