The main observation of this note is that the Lebesgue measure μ in the Turán-Nazarov inequality for exponential polynomials can be replaced with a certain geometric invariant ω ≥ μ, 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 ω.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm218-1-2,
author = {Omer Friedland and Yosef Yomdin},
title = {An observation on the Tur\'an-Nazarov inequality},
journal = {Studia Mathematica},
volume = {215},
year = {2013},
pages = {27-39},
zbl = {1292.26039},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm218-1-2}
}
Omer Friedland; Yosef Yomdin. An observation on the Turán-Nazarov inequality. Studia Mathematica, Tome 215 (2013) pp. 27-39. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm218-1-2/