Alan Haynes
University of York
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Publication
Featured researches published by Alan Haynes.
arXiv: Dynamical Systems | 2014
Alan Haynes; Michael Kelly; Barak Weiss
In 1998, Burago-Kleiner and McMullen independently proved the existence of separated nets in
Journal of Number Theory | 2003
Alan Haynes
\mathbb{R}^d
Israel Journal of Mathematics | 2016
Alan Haynes; Henna Koivusalo
which are not bi-Lipschitz equivalent (BL) to a lattice. A finer equivalence relation than BL is bounded displacement (BD). Separated nets arise naturally as return times to a section for minimal
Ergodic Theory and Dynamical Systems | 2016
Alan Haynes
\mathbb{R}^d
arXiv: Dynamical Systems | 2016
Alan Haynes; Henna Koivusalo; James Walton; Lorenzo Sadun
-actions. We analyze the separated nets which arise via these constructions, focusing particularly on nets arising from linear
American Mathematical Monthly | 2015
Alan Haynes; Sara Munday
\mathbb{R}^d
International Journal of Number Theory | 2010
Alan Haynes
-actions on tori. We show that generically these nets are BL to a lattice, and for some choices of dimensions and sections, they are generically BD to a lattice. We also show the existence of such nets which are not BD to a lattice.
arXiv: Number Theory | 2008
Alan Haynes; Kosuke Homma
Abstract In this paper we examine the subset F Q, odd of Farey fractions of order Q consisting of those fractions whose denominators are odd. In particular, we consider the frequencies of the values taken on by Δ=qa′−aq′ where a/q F Q, odd . After proving an asymptotic result for these frequencies, we generalize the result to the subset of elements of F Q, odd formed by restriction to a subinterval [α,β]⊆[0,1].
Symmetry Integrability and Geometry-methods and Applications | 2015
Alan Haynes; Wadim Zudilin
For any irrational cut-and-project setup, we demonstrate a natural infinite family of windows which gives rise to separated nets that are each bounded distance to a lattice. Our proof provides a new construction, using a sufficient condition of Rauzy, of an infinite family of non-trivial bounded remainder sets for any totally irrational toral rotation in any dimension.
International Journal of Number Theory | 2013
Tim D Browning; Alan Haynes
We prove that in any totally irrational cut-and-project setup with codimension (internal space dimension) one, it is possible to choose sections (windows) in non-trivial ways so that the resulting sets are bounded displacement equivalent to lattices. Our proof demonstrates that for any irrational