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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 <i>V</i> with an arbitrarily small Lebesgue measure such that for any positive integer <i>N</i>, the set of exponentials with frequencies in any union of cosets of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mi mathvariant="double-struck">Z</mi></mrow></semantics></math></inline-formula> cannot be a frame for the space of square integrable functions over <i>V</i>. These results are based on the proof technique of Olevskii and Ulanovskii from 2008.