Prime and almost prime integral points on principal homogeneous spaces

Amos Nevo Technion – Israel Institute of Technology, Technion City, Haifa, Israel Peter Sarnak School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ, U.S.A.

TBD mathscidoc:1701.332022

Acta Mathematica, 205, (2), 361-402, 2008.11
We develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space. In particular, we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers. When the orbit is the set of integral matrices of a fixed determinant, we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups, and sharp and uniform counting of points on such orbits when ordered by various norms.
affine sieve; semisimple groups; arithmetic lattices; lattice points; prime numbers; principal homogeneous spaces; spectral gap; mean ergodic theorem
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  title={Prime and almost prime integral points on principal homogeneous spaces},
  author={Amos Nevo, and Peter Sarnak},
  booktitle={Acta Mathematica},
Amos Nevo, and Peter Sarnak. Prime and almost prime integral points on principal homogeneous spaces. 2008. Vol. 205. In Acta Mathematica. pp.361-402.
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