Affine linear sieve, expanders, and sum-product
Tóm tắt
Let
$\mathcal{O}$
be an orbit in ℤ
n
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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