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Gaps problems and frequencies of patches in cut and project sets

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JournalMathematical Proceedings of the Cambridge Philosophical Society
DateE-pub ahead of print - 3 Mar 2016
DatePublished (current) - Jul 2016
Issue number1
Volume161
Pages (from-to)65-85
Early online date3/03/16
Original languageEnglish

Abstract

We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets of full Hausdorff dimension.

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