Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

Request Permissions   Purchase Content 


The affine sieve

Authors: Alireza Salehi Golsefidy and Peter Sarnak
Journal: J. Amer. Math. Soc. 26 (2013), 1085-1105
MSC (2010): Primary 20G35, 11N35
Published electronically: April 1, 2013
MathSciNet review: 3073885
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802 (39 #4129)
  • [BW93] 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 (95e:13018)
  • [BG08] Jean Bourgain and Alex Gamburd, Uniform expansion bounds for Cayley graphs of $ {\rm SL}_2(\mathbb{F}_p)$, Ann. of Math. (2) 167 (2008), no. 2, 625-642. MR 2415383 (2010b:20070),
  • [BGS10] Jean Bourgain, Alex Gamburd, and Peter Sarnak, Affine linear sieve, expanders, and sum-product, Invent. Math. 179 (2010), no. 3, 559-644. MR 2587341 (2011d:11018),
  • [BGT11] Emmanuel Breuillard, Ben Green, and Terence Tao, Approximate subgroups of linear groups, Geom. Funct. Anal. 21 (2011), no. 4, 774-819. MR 2827010,
  • [BLMS05] 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 (2005k:11020)
  • [Ch73] 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 0434997 (55 #7959)
  • [FHM94] 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 (95k:12016),
  • [G69] 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 0217086 (36 #178)
  • [HR74] H. Halberstam and H.-E. Richert, Sieve methods, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974. London Mathematical Society Monographs, No. 4. MR 0424730 (54 #12689)
  • [HW79] 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 (81i:10002)
  • [H08] H. A. Helfgott, Growth and generation in $ {\rm SL}_2(\mathbb{Z}/p\mathbb{Z})$, Ann. of Math. (2) 167 (2008), no. 2, 601-623. MR 2415382 (2009i:20094),
  • [LW54] Serge Lang and André Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819-827. MR 0065218 (16,398d)
  • [M55] G. D. Mostow, Self-adjoint groups, Ann. of Math. (2) 62 (1955), 44-55. MR 0069830 (16,1088a)
  • [NS10] 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 (2011m:22040),
  • [N87] Madhav V. Nori, On subgroups of $ {\rm GL}_n({\bf F}_p)$, Invent. Math. 88 (1987), no. 2, 257-275. MR 880952 (88d:20068),
  • [PR94] 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 (95b:11039)
  • [PS] L. Pyber and E. Szabó, Growth in finite simple groups of Lie type of bounded rank, preprint.
  • [Ra72] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, New York, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68. MR 0507234 (58 #22394a)
  • [SV] Alireza Salehi Golsefidy and Péter P. Varjú, Expansion in perfect groups, Geom. Funct. Anal. 22 (2012), no. 6, 1832-1891. MR 3000503,
  • [Sch00] 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 (2001h:11135)
  • [Sc74] Wolfgang M. Schmidt, A lower bound for the number of solutions of equations over finite fields, J. Number Theory 6 (1974), 448-480. Collection of articles dedicated to K. Mahler on the occasion of his seventieth birthday. MR 0360598 (50 #13045)
  • [Sp98] T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston Inc., Boston, MA, 1998. MR 1642713 (99h:20075)
  • [V12] Péter P. Varjú, Expansion in $ SL_d(\mathcal {O}_K/I)$, $ I$ square-free, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 1, 273-305. MR 2862040,

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 20G35, 11N35

Retrieve articles in all journals with MSC (2010): 20G35, 11N35

Additional Information

Alireza Salehi Golsefidy
Affiliation: Department of Mathematics, University of California, San Diego, California 92093-0112

Peter Sarnak
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000

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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society