### The inverse sieve problem in high dimensions

Miguel N. Walsh
Source: Duke Math. J. Volume 161, Number 10 (2012), 2001-2022.

#### Abstract

We show that if a big set of integer points $S\subseteq[0,N]^{d}$, $d\textgreater 1$, occupies few residue classes mod $p$ for many primes $p$, then it must essentially lie in the solution set of some polynomial equation of low degree. This answers a question of Helfgott and Venkatesh.

First Page:
Primary Subjects: 11N35
Secondary Subjects: 11B30, 11N69
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1340801630
Digital Object Identifier: doi:10.1215/00127094-1645788
Zentralblatt MATH identifier: 06063227
Mathematical Reviews number (MathSciNet): MR2954623

### References

[1] E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), 337–357.
Mathematical Reviews (MathSciNet): MR1016893
Zentralblatt MATH: 0718.11048
Digital Object Identifier: doi:10.1215/S0012-7094-89-05915-2
Project Euclid: euclid.dmj/1077308005
[2] A. C. Cojocaru and M. R. Murty, An Introduction to Sieve Methods and Their Applications, London Math. Soc. Stud. Texts 66, Cambridge Univ. Press, Cambridge, 2006.
Mathematical Reviews (MathSciNet): MR2200366
Zentralblatt MATH: 1121.11063
[3] E. Croot and V. F. Lev, “Open problems in additive combinatorics” in Additive Combinatorics, CRM Proc. Lecture Notes 43, Amer. Math. Soc., Providence, 2007, 207–233.
Mathematical Reviews (MathSciNet): MR2359473
Zentralblatt MATH: 1183.11005
[4] P. X. Gallagher, A larger sieve, Acta Arith. 18 (1971), 77–81.
Mathematical Reviews (MathSciNet): MR291120
[5] B. J. Green, Approximate groups and their applications: Work of Bourgain, Gamburd, Helfgott and Sarnak, preprint, arXiv:0911.3354v2 [math.NT]
arXiv: 0911.3354v2
[6] B. J. Green, On a variant of the large sieve, preprint, arXiv:0807.5037v2 [math.NT]
arXiv: 0807.5037v2
[7] B. J. Green and I. Z. Ruzsa, Freiman’s theorem in an arbitrary abelian group, J. Lond. Math. Soc. (2) 75 (2007), 163–175.
Mathematical Reviews (MathSciNet): MR2302736
Zentralblatt MATH: 1133.11058
Digital Object Identifier: doi:10.1112/jlms/jdl021
[8] B. J. Green, T. Tao, and T. Ziegler, An inverse theorem for the Gowers $U^{s+1}[N]$-norm, to appear in Ann. of Math. (2), preprint, arXiv:1009.3998v3 [math.CO]
arXiv: 1009.3998v3
[9] R. R. Hall, On pseudo-polynomials, Mathematika 18 (1971), 71–77.
Mathematical Reviews (MathSciNet): MR294300
Digital Object Identifier: doi:10.1112/S002557930000838X
[10] H. A. Helfgott, Growth and generation in $\mathrm{SL}_{2}(\mathbb{Z}/p\mathbb{Z})$, Ann. of Math. (2) 167 (2008), 601–623.
Mathematical Reviews (MathSciNet): MR2415382
Zentralblatt MATH: 1213.20045
Digital Object Identifier: doi:10.4007/annals.2008.167.601
[11] H. A. Helfgott and A. Venkatesh, “How small must ill-distributed sets be?” in Analytic Number Theory: Essays in Honour of Klaus Roth, Cambridge Univ. Press, Cambridge, 2009, 224–234.
Mathematical Reviews (MathSciNet): MR2508647
Zentralblatt MATH: 1217.11073
[12] E. Kowalski, The Large Sieve and Its Applications: Arithmetic Geometry, Random Walks and Discrete Groups, Cambridge Tracts in Math. 175, Cambridge Univ. Press, Cambridge, 2008.
Mathematical Reviews (MathSciNet): MR2426239
Zentralblatt MATH: 1177.11080
[13] E. Kowalski, The ubiquity of surjective reduction in random groups, unpublished notes, October 2007, http://www.math.ethz.ch/~kowalski/notes-unpublished.html.
[14] H. L. Montgomery, A note on the large sieve, J. London Math. Soc. 43 (1968), 93–98.
Mathematical Reviews (MathSciNet): MR224585
Digital Object Identifier: doi:10.1112/jlms/s1-43.1.93
[15] I. Ruzsa, Jr., On congruence preserving functions, Mat. Lapok. 22 (1971), 125–134.
Mathematical Reviews (MathSciNet): MR323688
[16] J. Sándor, D. S. Mitrinovic, and B. Crstici, Handbook of Number Theory, I, Springer, Dordrecht, 2006.
Mathematical Reviews (MathSciNet): MR2186914
[17] T. Tao, Freiman’s theorem for solvable groups, Contrib. Discrete Math. 5 (2010), 137–184.
Mathematical Reviews (MathSciNet): MR2791295
[18] T. Tao and V. Vu, Additive Combinatorics, Cambridge Stud. Adv. Math. 105, Cambridge Univ. Press, Cambridge, 2006.
Mathematical Reviews (MathSciNet): MR2289012
[19] T. Tao and V. Vu, Inverse Littlewood-Offord theorems and the condition number of random matrices, Ann. of Math. (2) 169 (2009), 595–632.
Mathematical Reviews (MathSciNet): MR2480613
Zentralblatt MATH: 05578752
Digital Object Identifier: doi:10.4007/annals.2009.169.595