The Wiener Lemma and Cocycles

Abstract

We give a sufficient and necessary condition for a function with its values in the unit circle to be a multiplicative coboundary. This theorem generalizes the following result of Veech [1]. Let $T:{\mathbf {T}} \to {\mathbf {T}}$ be a rotation of the unit circle ${\mathbf {T}}$ by an irrational angle $\theta$. Let $F:{\mathbf {T}} \to {\mathbf {T}}$ be a measurable function. Then $F$ is a multiplicative coboundary iff \[ \int _{\mathbf {T}} {F(x)F(Tx) \cdots F({T^{n - 1}}x)d\mu (x) \to 1,\quad {\text {as }}\left \| {n\theta } \right \| \to 0,} \] where $\left \| {n\theta } \right \|$ is the distance of $n\theta$ from integers and $\mu$ is the Haar measure.