For d >= 4, primitive rank-(d-1) subgroups of Z^d with fixed covolume T become equidistributed toward a rotation-invariant 'polar' measure as T grows along admissible subsequences, extending Aka-Einsiedler-Shapira.
Linnik's ergodic method and the distribution of integer points on spheres
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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).
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On the distribution of primitive subgroups of $\mathbb{Z}^{d}$ of large covolume
For d >= 4, primitive rank-(d-1) subgroups of Z^d with fixed covolume T become equidistributed toward a rotation-invariant 'polar' measure as T grows along admissible subsequences, extending Aka-Einsiedler-Shapira.