Affine linear sieve, expanders, and sum-product
The paper under review is an expanded version of the authors' announcement in [C. R., Math., Acad. Sci. Paris 343, No. 3, 155--159 (2006; Zbl 1217.11081)]. From the abstract: Let \(\mathcal{O}\) be an orbit in \(\mathbb Z^{n}\) of a finitely generated subgroup \(\Lambda \) of \(\text{GL}_{n }(\mathbb Z)\) whose Zariski closure \(\text{Zcl}(\Lambda )\) is suitably large (e.g. isomorphic to \(\text{SL}_{2}\)). 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 \(\text{Zcl}(\Lambda )=\,\text{SL}_{2}\), using crucially sum-product theorem in \(\mathbb Z/q\mathbb Z\) for \(q\) square-free. For a detailed survey of the context, scope and significance of these results we refer the reader to the first chapter of (\url{http://www.ams.org/meetings/lectures/currentevents2010.pdf}) by B. Green.
scientific article; zbMATH DE number 5675041
scientific article