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Gaps problems and frequencies of patches in cut and project sets

Published online by Cambridge University Press:  03 March 2016

ALAN HAYNES ,
HENNA KOIVUSALO ,
JAMES WALTON  and
LORENZO SADUN
ALAN HAYNES
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mails: alan.haynes@york.ac.uk; henna.koivusalo@york.ac.uk; jamie.walton@york.ac.uk
HENNA KOIVUSALO
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mails: alan.haynes@york.ac.uk; henna.koivusalo@york.ac.uk; jamie.walton@york.ac.uk
JAMES WALTON
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD. e-mails: alan.haynes@york.ac.uk; henna.koivusalo@york.ac.uk; jamie.walton@york.ac.uk
LORENZO SADUN
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, TX 78712-1082, U.S.A. e-mail: sadun@math.utexas.edu
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 1 , July 2016 , pp. 65 - 85
Copyright
Copyright © Cambridge Philosophical Society 2016 

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