Mathematics Colloquia and Seminars

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For a fixed vector w in R^2, consider the following increasing sequence of subsets of the unit circle: {(v+w)/|v+w| where v is in Z^2 and |v+w|<N} and N tends to infinity. This represents the directions of points in the lattice w + Z^2. There have been various results on the distribution of gaps of these points on the circle: for example, when the coordinates of w are irrational, the gap distribution of this sequence was proved by Marklof and Strombergsson to coincide with the gap distribution of the sequence (sqrt{n} mod 1). A hyperbolic version of this Euclidean problem, in which the lattice Z^2 is replaced by a lattice (a subgroup of finite covolume) in PSL_2(R), was considered recently by Marklof and Vinogradov. In our talk, we take this hyperbolic problem one step further and replace the lattice by a subgroup of PSL_2(R) of infinite covolume.