## The affine sieve

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- by Alireza Salehi Golsefidy and Peter Sarnak
- J. Amer. Math. Soc.
**26**(2013), 1085-1105 - DOI: https://doi.org/10.1090/S0894-0347-2013-00764-X
- Published electronically: April 1, 2013
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## Abstract:

We establish the main saturation conjecture connected with executing a Brun sieve in the setting of an orbit of a group of affine linear transformations. This is carried out under the condition that the Zariski closure of the group is Levi-semisimple. It is likely that this condition is also necessary for such saturation to hold.## References

- M. F. Atiyah and I. G. Macdonald,
*Introduction to commutative algebra*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR**0242802** - Thomas Becker and Volker Weispfenning,
*Gröbner bases*, Graduate Texts in Mathematics, vol. 141, Springer-Verlag, New York, 1993. A computational approach to commutative algebra; In cooperation with Heinz Kredel. MR**1213453**, DOI 10.1007/978-1-4612-0913-3 - Jean Bourgain and Alex Gamburd,
*Uniform expansion bounds for Cayley graphs of $\textrm {SL}_2(\Bbb F_p)$*, Ann. of Math. (2)**167**(2008), no. 2, 625–642. MR**2415383**, DOI 10.4007/annals.2008.167.625 - Jean Bourgain, Alex Gamburd, and Peter Sarnak,
*Affine linear sieve, expanders, and sum-product*, Invent. Math.**179**(2010), no. 3, 559–644. MR**2587341**, DOI 10.1007/s00222-009-0225-3 - Emmanuel Breuillard, Ben Green, and Terence Tao,
*Approximate subgroups of linear groups*, Geom. Funct. Anal.**21**(2011), no. 4, 774–819. MR**2827010**, DOI 10.1007/s00039-011-0122-y - Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek,
*On Fibonacci numbers with few prime divisors*, Proc. Japan Acad. Ser. A Math. Sci.**81**(2005), no. 2, 17–20. MR**2126070** - Jing Run Chen,
*On the representation of a larger even integer as the sum of a prime and the product of at most two primes*, Sci. Sinica**16**(1973), 157–176. MR**434997** - Michael D. Fried, Dan Haran, and Moshe Jarden,
*Effective counting of the points of definable sets over finite fields*, Israel J. Math.**85**(1994), no. 1-3, 103–133. MR**1264342**, DOI 10.1007/BF02758639 - A. Grothendieck,
*Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III*, Inst. Hautes Études Sci. Publ. Math.**28**(1966), 255. MR**217086** - H. Halberstam and H.-E. Richert,
*Sieve methods*, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR**0424730** - G. H. Hardy and E. M. Wright,
*An introduction to the theory of numbers*, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR**568909** - H. A. Helfgott,
*Growth and generation in $\textrm {SL}_2(\Bbb Z/p\Bbb Z)$*, Ann. of Math. (2)**167**(2008), no. 2, 601–623. MR**2415382**, DOI 10.4007/annals.2008.167.601 - Serge Lang and André Weil,
*Number of points of varieties in finite fields*, Amer. J. Math.**76**(1954), 819–827. MR**65218**, DOI 10.2307/2372655 - G. D. Mostow,
*Self-adjoint groups*, Ann. of Math. (2)**62**(1955), 44–55. MR**69830**, DOI 10.2307/2007099 - Amos Nevo and Peter Sarnak,
*Prime and almost prime integral points on principal homogeneous spaces*, Acta Math.**205**(2010), no. 2, 361–402. MR**2746350**, DOI 10.1007/s11511-010-0057-4 - Madhav V. Nori,
*On subgroups of $\textrm {GL}_n(\textbf {F}_p)$*, Invent. Math.**88**(1987), no. 2, 257–275. MR**880952**, DOI 10.1007/BF01388909 - Vladimir Platonov and Andrei Rapinchuk,
*Algebraic groups and number theory*, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR**1278263** - L. Pyber and E. Szabó,
*Growth in finite simple groups of Lie type of bounded rank,*preprint. - M. S. Raghunathan,
*Discrete subgroups of Lie groups*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR**0507234** - A. Salehi Golsefidy and Péter P. Varjú,
*Expansion in perfect groups*, Geom. Funct. Anal.**22**(2012), no. 6, 1832–1891. MR**3000503**, DOI 10.1007/s00039-012-0190-7 - A. Schinzel,
*Polynomials with special regard to reducibility*, Encyclopedia of Mathematics and its Applications, vol. 77, Cambridge University Press, Cambridge, 2000. With an appendix by Umberto Zannier. MR**1770638**, DOI 10.1017/CBO9780511542916 - Wolfgang M. Schmidt,
*A lower bound for the number of solutions of equations over finite fields*, J. Number Theory**6**(1974), 448–480. MR**360598**, DOI 10.1016/0022-314X(74)90043-2 - T. A. Springer,
*Linear algebraic groups*, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR**1642713**, DOI 10.1007/978-0-8176-4840-4 - Péter P. Varjú,
*Expansion in $SL_d(\scr O_K/I)$, $I$ square-free*, J. Eur. Math. Soc. (JEMS)**14**(2012), no. 1, 273–305. MR**2862040**, DOI 10.4171/JEMS/302

## Bibliographic Information

**Alireza Salehi Golsefidy**- Affiliation: Department of Mathematics, University of California, San Diego, California 92093-0112
- Email: golsefidy@ucsd.edu
**Peter Sarnak**- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
- MR Author ID: 154725
- Email: sarnak@math.princeton.edu
- Received by editor(s): October 13, 2011
- Received by editor(s) in revised form: January 7, 2013
- Published electronically: April 1, 2013
- Additional Notes: The first author was partially supported by the NSF grants DMS-0635607 and DMS-1001598

The second author was partially supported by an NSF grant - © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**26**(2013), 1085-1105 - MSC (2010): Primary 20G35, 11N35
- DOI: https://doi.org/10.1090/S0894-0347-2013-00764-X
- MathSciNet review: 3073885