Density of Zeros of the Cartwright Class Functions and the Helson–Szegő Type Condition

B. Ya. Levin has proved that the zero set of a sine type function can be represented as a union of finitely many separated sets, which 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 $\sup|{\operatorname{Im}\lambda_n}|<\infty$ such that $|F(x)|^2$ satisfies the Helson–Szegő condition. Moreover, we show that instead of the last condition one can require that $\log|F(x)|$ belongs to the BMO class.