On Construction of Bounded Sets Not Admitting a General Type of Riesz Spectrum

Abstract

We construct a bound set that does not admit a Riesz spectrum containing a nonempty periodic set for which the period is a rational multiple of a fixed constant. As a consequence, we obtain a bounded set V with an arbitrarily small Lebesgue measure such that for any positive integer N, the set of exponentials with frequencies in any union of cosets of NZ cannot be a frame for the space of square integrable functions over V. These results are based on the proof technique of Olevskii and Ulanovskii from 2008.

Keywords

• complex exponentials
• spectrum
• exponential bases
• Riesz bases
• Riesz sequences
• frames