Density of Zeros of the Cartwright Class Functions and the Helson--Szeg\H{o} Type Condition

S. A. Avdonin ; S. A. Ivanov

Matematičeskie zametki, Tome 113 (2023) no. 2, pp. 163-170.

Résumé

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.

Détail

Keywords: Helson–Szegő condition, upper uniform density, exponential Riesz bases.

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