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

Abstract

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).

References in corpus (2)

• The spectral decomposition of shifted convolution sums
• Local-global principles for representations of quadratic forms

Cited by in corpus (11)

• Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics
• Complexity of strong approximation on the sphere
• p-Adic distribution of CM points and Hecke orbits. II: Linnik equidistribution on the supersingular locus
• Spatial statistics for lattice points on the sphere I: Individual results
• Equidistribution of primitive vectors, and the shortest solutions to their GCD equations
• The Siegel variance formula for quadratic forms
• Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbits
• Equidistribution of primitive lattices in ${\mathbb{R}}^n$
• Expander Graphs in Pure and Applied Mathematics
• Large Deviations and Effective Equidistribution
• On the distribution of primitive subgroups of ${\mathbb{Z}}^d$ of large covolume