Gaps problems and frequencies of patches in cut and project sets

Haynes, A., Koivusalo, H., Walton, J. and Sadun, L. (2016) Gaps problems and frequencies of patches in cut and project sets. Mathematical Proceedings of the Cambridge Philosophical Society, 161(1), pp. 65-85. (doi: 10.1017/S0305004116000128)

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.

• Item Type: Articles
• Additional Information: AH, HK, JW: Research supported by EPSRC. LS: Research partially supported by NSF Grant DMS-1101326.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Walton, Dr Jamie
• Authors: Haynes, A., Koivusalo, H., Walton, J., and Sadun, L.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Mathematical Proceedings of the Cambridge Philosophical Society
• Publisher: Cambridge University Press
• ISSN: 0305-0041
• ISSN (Online): 1469-8064
• Published Online: 03 March 2016
• Copyright Holders: Copyright © 2016 Cambridge Philosophical Society
• First Published: First published in Mathematical Proceedings of the Cambridge Philosophical Society 161(1): 65-85
• Publisher Policy: Reproduced in accordance with the publisher copyright policy

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