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

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).