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Affine linear sieve, expanders, and sum-product. (English) Zbl 1239.11103

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 (http://www.ams.org/meetings/lectures/currentevents2010.pdf) by B. Green.

MSC:

11N35 Sieves
11B30 Arithmetic combinatorics; higher degree uniformity
11B13 Additive bases, including sumsets

Citations:

Zbl 1217.11081
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Full Text: DOI

References:

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