Abstract: Duke's theorem from the late 80's describes the asymptotic
distribution of closed geodesics on the modular surface and similar
problems. Earlier work by Linnik on the same problem used an ergodic
method later reinterpretated in a modern setting by Einsiedler,
Lindenstrauss, Michel, and Venkatesh. A crucial ingredient to Linnik's
ergodic method is an estimate known as Linnik's basic lemma which
controls the measure of nearby points.

In joint work with Pengyu Yang, we give a new proof of Linnik's basic
lemma. Our method allows us to give a version of Linnik's basic lemma
over number fields with (optimal) uniformity over the base field. The
goal of this talk will be to explain Linnik's basic lemma as well as the
new proof (over the rational field) from first principles.


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