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Points on spheres and their orthogonal lattices

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Interactions between Dynamics of Group Actions and Number Theory

It is a classical question to understand the distribution (when projected to the unit sphere) of the solutions of x2+y2+z^2=D as D grows. To each such solution v we further attach the lattice obtained by intersecting the hyperplane orthogonal to v with the set of integral vectors. This way, we obtain, for any D that can be written as a sum of three squares, a finite set of pairs consisting of a point on the unit sphere and a lattice. In the talk I will discuss a joint work with Manfred Einsiedler and Uri Shapira which considers the joint distribution of these pairs in the appropriate spaces. I will outline a general approach to such problems and discuss dynamical input needed to establish that these pairs distribute uniformly.

This talk is part of the Isaac Newton Institute Seminar Series series.

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