An observation on the Turán-Nazarov inequality

Abstract

The main observation of this note is that the Lebesgue measure $\mu$ in the Turán-Nazarov inequality for exponential polynomials can be replaced with a certain geometric invariant $\omega \ge \mu$, which can be effectively estimated in terms of the metric entropy of a set, and may be nonzero for discrete and even finite sets. While the frequencies (the imaginary parts of the exponents) do not enter in the original Turán-Nazarov inequality, they necessarily enter the definition of $\omega$.