The affine sieve

Salehi Golsefidy, Alireza; Sarnak, Peter

doi:10.1090/S0894-0347-2013-00764-X

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