Given a set of positive measure on the circle and a set Λ of integers, one can ask whether is a Riesz sequence in L²(). We consider this question in connection with some arithmetic properties of the set Λ. Improving a result of Bownik and Speegle (2006), we construct a set such that E(Λ) is never a Riesz sequence if Λ contains an arithmetic progression of length N and step with N arbitrarily large. On the other hand, we prove that every set admits a Riesz sequence E(Λ) such that Λ does contain arithmetic progressions of length N and step ℓ = O(N) with N arbitrarily large.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm225-2-5,
author = {Itay Londner and Alexander Olevski\u\i },
title = {Riesz sequences and arithmetic progressions},
journal = {Studia Mathematica},
volume = {223},
year = {2014},
pages = {183-191},
zbl = {1322.42037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm225-2-5}
}
Itay Londner; Alexander Olevskiĭ. Riesz sequences and arithmetic progressions. Studia Mathematica, Tome 223 (2014) pp. 183-191. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm225-2-5/