Type alternative for Frostman measures

Abstract

For a finite positive Borel measure $\mu$ on $\mathbb R$ its exponential type, $T_\mu$, is defined as the infimum of $a>0$ such that finite linear combinations of complex exponentials with frequencies between 0 and $a$ are dense in $L^2(\mu)$. The definition can be easily extended from finite to broader classes of measures. In this paper we prove a new formula for $T_\mu$ and use it to study growth and additivity properties of measures with finite positive type. As one of the applications, we show that Frostman measures on $\mathbb R$ may only have type zero or infinity.