On a property of Herglotz functions

TL;DR

This paper proves a new property of Herglotz functions, showing that a certain measure derived from them has an absolutely continuous density that is almost everywhere either 0 or 1, revealing a surprising structure.

Contribution

It introduces a novel property of Herglotz functions relating to the measure's singular part and its density being integer-valued almost everywhere.

Findings

The measure rf3db3n is absolutely continuous.

The density of this measure is integer-valued almost everywhere.

The density takes values only 0 or 1 almost everywhere.

Abstract

In this note I prove the following property of Herglotz functions, which to my knowledge is new: For a Herglotz function $h(z)$ and a real number $r \in \mathbb R$ define a Herglotz function $g_r(z) = (r - h(z))^{-1}.$ Let $\mu_r^{(s)}$ be the singular part of the measure $\mu_r$ which corresponds to $g_r(z)$ via the Herglotz representation theorem. Then the measure $\int_0^1 \mu_r^{(s)}\,dr$ is absolutely continuous, its density is integer-valued a.e., and moreover the density takes values $0$ or $1$ a.e.