Type alternative for Frostman measures

TL;DR

This paper introduces a new formula for the exponential type of measures on the real line, explores its properties, and demonstrates that Frostman measures can only have type zero or infinity.

Contribution

It provides a novel formula for the exponential type of measures and investigates the growth and additivity properties, revealing that Frostman measures have only trivial types.

Findings

New formula for exponential type of measures

Frostman measures have only type zero or infinity

Insights into growth and additivity properties of 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.