Kloosterman Sums, Disjointness, and Equidistribution

Abstract

Kloosterman sums play a prominent role in number theory, in particular in the spectral theory of automorphic forms. In these notes, we relate Kloosterman sums to equidistribution properties of sparse subsets of horocycle orbits inside the homogeneous space \(X_{2}=\mathrm {SL}_{2}({{\mathbb {Z}}})\backslash \mathrm {SL}_{2}({\mathbb {R}})\). Moreover, we give a connection between Kloosterman sums and a disjointness result on the torus. The equidistribution inside X2 is proven using effective mixing of the \(\mathrm {SL}_{2}({\mathbb {R}})\)-action on X2, which relies on Kloosterman sums. Both discussions combine input of number theoretic flavor with purely dynamical arguments.