Fourier asymptotics of Cantor type measures at infinity

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.