Linnik's ergodic method and the distribution of integer points on spheres

Details Linnik's ergodic method and the distribution of integer points on spheres Linnik's ergodic method and the distribution of integer points on spheres

2010-01-06

arxiv.org/abs/1001.0897v1

Mathematics

Number Theory

Scientific paper

We discuss Linnik's work on the distribution of integral solutions to $x^2+y^2+z^2 =d$, as $d$ goes to infinity. We give an exposition of Linnik's ergodic method; indeed, by using large-deviation results for random walks on expander graphs, we establish a refinement of his equidistribution theorem. We discuss the connection of these ideas with modern developments (ergodic theory on homogeneous spaces, $L$-functions).

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