Fourier asymptotics of Cantor type measures at infinity

Abstract:

Let $q\geq 3$ be an integer and let $\phi (t)=\prod _{n=1}^\infty \cos (q^{-n}t)$. In this note we prove that $\lim _{t\to \infty } \phi (t)=-\phi (\pi )$ for all $q$; $\varlimsup _{t\to \infty }\phi (t)=\phi (\pi )$ if $q$ is odd and $\varlimsup _{t\to \infty }\phi (t)\le \phi (\pi )$ if $q$ is even$.$ This improves a classical result of Wiener and Wintner. We also give a necessary and sufficient condition for the product $\prod _{i=1}^m\phi (\alpha _it)$ to approach zero at infinity.