A ZERO- LAW FOR COSINE FAMILIES

Abstract

Let $a\in \mathbb{R}$, and let $k(a)$ be the largest constant such that $\sup |\text{cos}(na)-\cos (nb)|<k(a)$ for $b\in \mathbb{R}$ implies that $b\in \pm a+2\unicode[STIX]{x1D70B}\mathbb{Z}$. We show that if a cosine sequence $(C(n))_{n\in \mathbb{Z}}$ with values in a Banach algebra $A$ satisfies $\sup _{n\geq 1}\Vert C(n)-\cos (na).1_{A}\Vert <k(a)$, then $C(n)=\cos (na).1_{A}$ for $n\in \mathbb{Z}$. Since $\!\sqrt{5}/2\leq k(a)\leq 8/3\!\sqrt{3}$ for every $a\in \mathbb{R}$, this shows that if some cosine family $(C(g))_{g\in G}$ over an abelian group $G$ in a Banach algebra satisfies $\sup _{g\in G}\Vert C(g)-c(g)\Vert <\!\sqrt{5}/2$ for some scalar cosine family $(c(g))_{g\in G}$, then $C(g)=c(g)$ for $g\in G$, and the constant $\!\sqrt{5}/2$ is optimal. We also describe the set of all real numbers $a\in [0,\unicode[STIX]{x1D70B}]$ satisfying $k(a)\leq \frac{3}{2}$.