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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2213))

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 X 2 is proven using effective mixing of the \(\mathrm {SL}_{2}({\mathbb {R}})\)-action on X 2, which relies on Kloosterman sums. Both discussions combine input of number theoretic flavor with purely dynamical arguments.

This article summarizes the minicourse given by the first named author during the research school “Applications of Ergodic Theory in Number Theory” organized by Mariusz Lemańczyk and Sébastien Ferenczi at the CIRM in Marseille, Luminy, running October 17 to October 21, 2016.

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Notes

  1. 1.

    Here and in what follows matrix entries which are 0 are omitted.

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Acknowledgements

We thank Shahar Mozes, Cagri Sert, Uri Shapira, and Andreas Wieser for helpful comments on an earlier draft of this article. M.E. has learned a lot from Akshay Venkatesh and wants to thank him for many discussions related to these topics.

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Correspondence to M. Luethi .

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Einsiedler, M., Luethi, M. (2018). Kloosterman Sums, Disjointness, and Equidistribution. In: Ferenczi, S., Kułaga-Przymus, J., Lemańczyk, M. (eds) Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol 2213. Springer, Cham. https://doi.org/10.1007/978-3-319-74908-2_10

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