Prime and almost prime integral points on principal homogeneous spaces

By A. Nevo, Peter Sarnak. 2009

51 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a76937ba9b9e2aaa3e17417ae698dd8199d4d3e4

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Citations in the corpus

On the sup-norm of Maass cusp forms of large level (2010, methodology): “in [5], or with an approach via ergodic theory and decay of matrix coefficients as in Nevo–Sarnak [ 15 ] (for congruence subgroups).”
On representations of integers in thin subgroups of SL(2,Z) (2010, background): “ g2G; k<T jdˇ(X):f(g)j; 8 JEAN BOURGAIN AND ALEX KONTOROVICH that is, the supremal value of rst order derivatives of f in a ball of radius Tin G. The following is a form of \dualization" (cf. [NS09]), an analogue of Poisson summation. Theorem 2.1. Fix any 0 2 and a congruence subgroup 1(q) &”
Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, background): “17, as was done for a group variety in [48].”