Alan Haynes

Equivalence relations on separated nets arising from linear toral flows

In 1998, Burago-Kleiner and McMullen independently proved the existence of separated nets in

A note on Farey fractions with odd denominators

\mathbb{R}^d

Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices

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

Equivalence classes of codimension-one cut-and-project nets

\mathbb{R}^d

Gaps problems and frequencies of patches in cut and project sets

-actions. We analyze the separated nets which arise via these constructions, focusing particularly on nets arising from linear

DIOPHANTINE APPROXIMATION AND COLORING

\mathbb{R}^d

NUMERATORS OF DIFFERENCES OF NONCONSECUTIVE FAREY FRACTIONS

-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.

The group ring of

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].

\mathbb {Q}/\mathbb {Z}

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.

and an application of a divisor problem

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