Revista Matemática Iberoamericana

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Volume 29, Issue 2, 2013, pp. 739–747
DOI: 10.4171/RMI/737

Published online: 2013-04-22

Size of orthogonal sets of exponentials for the disk

Alex Iosevich[1] and Mihail N. Kolountzakis[2]

(1) University of Rochester, USA
(2) University of Crete, Iraklio, Greece

Suppose that $\Lambda \subseteq \mathbb{R}^2$ has the property that any two exponentials with frequency from $\Lambda$ are orthogonal in the space $L^2(D)$, where $D \subseteq \mathbb{R}^2$ is the unit disk. Such sets $\Lambda$ are known to be finite but it is not known if their size is uniformly bounded. We show that if there are two elements of $\Lambda$ which are distance $t$ apart then the size of $\Lambda$ is $O(t)$. As a consequence we improve a result of Iosevich and Jaming and show that $\Lambda$ has at most $O(R^{2/3})$ elements in any disk of radius $R$.

Keywords: Spectral sets, Fuglede’s conjecture

Iosevich Alex, Kolountzakis Mihail: Size of orthogonal sets of exponentials for the disk. Rev. Mat. Iberoamericana 29 (2013), 739-747. doi: 10.4171/RMI/737