Density of zeros of the Cartwright class functions and the Helson--Szeg\"o type condition

TL;DR

This paper extends Levin's result on zero sets of sine type functions to a broader class of entire functions with zeros in a strip, under Helson--Szeg"o or BMO conditions on their boundary behavior.

Contribution

It generalizes the structure of zero sets for entire functions beyond sine type functions, under Helson--Szeg"o and BMO conditions.

Findings

Zero set of the generalized functions can be decomposed into a finite union of separated sets.

The Helson--Szeg"o condition can be replaced by a BMO condition on log|F(x)|.

Extension of Levin's theorem to a wider class of entire functions with zeros in a strip.

Abstract

B.\,Ya.\,Levin has proved that zero set of a sine type function can be presented as a union of a finite number of separated sets, that is an important result in the theory of exponential Riesz bases. In the present paper we extend Levin's result to a more general class of entire functions $F(z)$ with zeros in a strip $0<q \leq \Im z \leq Q,$ such that $|F(x)|^2$ satisfies the Helson--Szeg\"o condition. Moreover, we demonstrate that instead of the last condition one can require that $\log|F(x)|$ belongs to the BMO class.