Alan Haynes
Gaps problems and frequencies of patches in cut and project sets
Haynes, Alan; Koivusalo, Henna; Sadun, Lorenzo; Walton, James
Authors
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 | Mar 28, 2024 |
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 .
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