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Integer points on spheres and their orthogonal lattices

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Abstract

Linnik proved in the late 1950’s the equidistribution of integer points on large spheres under a congruence condition. The congruence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec) using completely different techniques. We conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement. We use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition.

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Notes

  1. Strictly speaking this is not needed but slightly simplifies the argument in Sect. 4.1.1 (cf. the higher dimensionsal case in [1]).

  2. The argument from [1, Lemma 3.3] could also be used to prove primitivity without the assumption \(D\in {\mathbb {F}}\).

  3. A torsor of a group G is a set on which G acts freely and transitively.

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Acknowledgments

We would like to thank Elon Lindenstrauss, Philippe Michel, and Akshay Venkatesh for many fruitful conversations over the last years on various topics and research projects that lead to the current paper. While working on this project the authors visited the Israel Institute of Advanced Studies (IIAS) at the Hebrew University and its hospitality is deeply appreciated. We thank Peter Sarnak and Ruixiang Zhang for many conversations on these topics at the IIAS.

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Authors and Affiliations

  1. Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Menny Aka & Manfred Einsiedler

  2. Department of Mathematics, Technion, Haifa, Israel

    Uri Shapira

Authors
  1. Menny Aka
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  2. Manfred Einsiedler
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  3. Uri Shapira
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Corresponding author

Correspondence to Manfred Einsiedler.

Additional information

M.A. acknowledges the support of ISEF, Advanced Research Grant 228304 from the ERC, and SNF Grant 200021-152819. M.E. acknowledges the support of the SNF Grant 200021-127145 and 200021-152819. U.S. acknowledges the support of the Chaya fellowship and ISF Grant 357/13.

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Aka, M., Einsiedler, M. & Shapira, U. Integer points on spheres and their orthogonal lattices. Invent. math. 206, 379–396 (2016). https://doi.org/10.1007/s00222-016-0655-7

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