Arithmetic Study Group: Diophantine approximation and point patterns in cut and project sets
10 February 2015 14:00 in CM 103
A basic and important problem in number theory and dynamical systems is to understand the collection of return times to a given region of an irrational rotation of the circle. There is a natural generalization of this problem to higher dimensions, which leads to the study of higher dimensional point patterns called cut and project sets. In this talk we will discuss a connection between frequencies of patterns in cut and project sets, and gaps problems in Diophantine approximation. We will explain how the Diophantine approximation properties of the subspace defining a cut and project set can influence the number of possible frequencies of patterns of a given size. Once this connection is established, we will show how techniques from Diophantine approximation can be used to prove that the number of frequencies of patterns of size r, for a typical cut and project set, is almost always less than a power of log r. Furthermore, for a collection of cut and project sets of full Hausdorff dimension we can show that the number of frequencies of patterns of size r remains bounded as r tends to infinity. For comparison, the number of patterns of size r in any totally irrational cut and project set of dimension d always grows at least as fast as a constant times r^d.
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