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

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We discuss Linnik's work on the distribution of integral solutions to x²+y²+z² =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).