Gaps problems and frequencies of patches in cut and project sets

Haynes, Alan; Koivusalo, Henna; Sadun, Lorenzo; Walton, James

doi:10.1017/s0305004116000128

Gaps problems and frequencies of patches in cut and project sets

Haynes, Alan; Koivusalo, Henna; Sadun, Lorenzo; Walton, James

Authors

Alan Haynes

Henna Koivusalo

Lorenzo Sadun

James Walton

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.

Citation

Haynes, A., Koivusalo, H., Sadun, L., & Walton, J. (2016). Gaps problems and frequencies of patches in cut and project sets. Mathematical Proceedings of the Cambridge Philosophical Society, 161(01), 65-85. https://doi.org/10.1017/s0305004116000128

Journal Article Type	Article
Acceptance Date	Jan 14, 2016
Online Publication Date	Mar 3, 2016
Publication Date	Jul 1, 2016
Deposit Date	Feb 21, 2017
Publicly Available Date	Jul 4, 2017
Journal	Mathematical Proceedings of the Cambridge Philosophical Society
Print ISSN	0305-0041
Electronic ISSN	1469-8064
Publisher	Cambridge University Press
Peer Reviewed	Peer Reviewed
Volume	161
Issue	01
Pages	65-85
DOI	https://doi.org/10.1017/s0305004116000128

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Copyright Statement
This article has been published in a revised form in Mathematical Proceedings of the Cambridge Philosophical Society https://doi.org/10.1017/S0305004116000128. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © Cambridge Philosophical Society 2016 .