Abstract
Let \(\mathcal{O}\) be an orbit in ℤn of a finitely generated subgroup Λ of GL n (ℤ) whose Zariski closure Zcl(Λ) is suitably large (e.g. isomorphic to SL2). We develop a Brun combinatorial sieve for estimating the number of points on \(\mathcal{O}\) at which a fixed integral polynomial is prime or has few prime factors, and discuss applications to classical problems, including Pythagorean triangles and integral Apollonian packings. A fundamental role is played by the expansion property of the “congruence graphs” that we associate with \(\mathcal{O}\) . This expansion property is established when Zcl(Λ)=SL2, using crucially sum-product theorem in ℤ/qℤ for q square-free.
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The first author was supported in part by the NSF. The second author was supported in part by DARPA, the NSF and Sloan Foundation. The third author was supported in part by Veblen Fund (IAS) and the NSF.
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Bourgain, J., Gamburd, A. & Sarnak, P. Affine linear sieve, expanders, and sum-product. Invent. math. 179, 559–644 (2010). https://doi.org/10.1007/s00222-009-0225-3
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DOI: https://doi.org/10.1007/s00222-009-0225-3