Density of zeros of the Cartwright class functions and the Helson--Szegö type condition

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 $0<q\le \mathrm{\Im }z\le Q,$ such that $\mathrm{\mid }F(x){\mathrm{\mid }}^2$ satisfies the Helson–Szegö condition. Moreover, we demonstrate that instead of the last condition one can require that $\log \mathrm{\mid }F(x)\mathrm{\mid }$ belongs to the BMO class.