An observation on the Turán–Nazarov inequality
Volume 218 / 2013
Studia Mathematica 218 (2013), 27-39
MSC: Primary 26D05; Secondary 30E05, 42A05.
DOI: 10.4064/sm218-1-2
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 the original Turán–Nazarov inequality, they necessarily enter the definition of $\omega $.
