An observation on the Tur\'an-Nazarov inequality

TL;DR

This paper refines the Turán-Nazarov inequality for exponential polynomials by replacing Lebesgue measure with a geometric invariant that accounts for set complexity and can be nonzero for discrete sets.

Contribution

It introduces a new geometric invariant  that extends the inequality to discrete and finite sets, incorporating metric entropy.

Findings

The geometric invariant  can be effectively estimated via metric entropy.

 can be nonzero even for finite sets.

The frequencies influence the invariant , unlike in the original inequality.

Abstract

The main observation of this note is that the Lebesgue measure $\mu$ in the Tur\'an-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 Tur\'an-Nazarov inequality, they necessarily enter the definition of $\omega$.