Density of orbits of semigroups of endomorphisms acting on the Adeles

We investigate the question of whether or not the orbit of a point in A/Q, under the natural action of a subset S of Q, is dense in A/Q. We prove that if the set S is a multiplicative semigroup which contains at least two multiplicatively independent elements, one of which is an integer, then the orbit under S of any point with irrational real coordinate is dense.Comment: 13 page

Diophantine approximation and coloring

We demonstrate how connections between graph theory and Diophantine approximation can be used in conjunction to give simple and accessible proofs of seemingly difficult results in both subjects.Comment: 16 pages, pre-publication version of paper which will appear in American Mathematical Monthl

A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs

We study the two-dimensional continued fraction algorithm introduced in \cite{garr} and the associated \emph{triangle map} $T$, defined on a triangle $\triangle\subset \R^2$. We introduce a slow version of the triangle map, the map $S$, which is ergodic with respect to the Lebesgue measure and preserves an infinite Lebesgue-absolutely continuous invariant measure. We discuss the properties that the two maps $T$ and $S$ share with the classical Gauss and Farey maps on the interval, including an analogue of the weak law of large numbers and of Khinchin's weak law for the digits of the triangle sequence, the expansion associated to $T$. Finally, we confirm the role of the map $S$ as a two-dimensional version of the Farey map by introducing a complete tree of rational pairs, constructed using the inverse branches of $S$, in the same way as the Farey tree is generated by the Farey map, and then, equivalently, generated by a generalised mediant operation.Comment: 32 pages. The main results have slightly changed due to a mistake in the previous versio

Diophantine approximation and coloring

We demonstrate how connections between graph theory and Diophantine approximation can be used in conjunction to give simple and accessible proofs of seemingly difficult results in both subjects