Duke Mathematical Journal

The inverse sieve problem in high dimensions

Miguel N. Walsh

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We show that if a big set of integer points S[0,N]d, d>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.

Article information

Duke Math. J., Volume 161, Number 10 (2012), 2001-2022.

First available in Project Euclid: 27 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11N35: Sieves
Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity 11N69: Distribution of integers in special residue classes


Walsh, Miguel N. The inverse sieve problem in high dimensions. Duke Math. J. 161 (2012), no. 10, 2001--2022. doi:10.1215/00127094-1645788. https://projecteuclid.org/euclid.dmj/1340801630

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  • [1] E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), 337–357.
  • [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.
  • [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.
  • [4] P. X. Gallagher, A larger sieve, Acta Arith. 18 (1971), 77–81.
  • [5] B. J. Green, Approximate groups and their applications: Work of Bourgain, Gamburd, Helfgott and Sarnak, preprint, arXiv:0911.3354v2 [math.NT]
  • [6] B. J. Green, On a variant of the large sieve, preprint, arXiv:0807.5037v2 [math.NT]
  • [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.
  • [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]
  • [9] R. R. Hall, On pseudo-polynomials, Mathematika 18 (1971), 71–77.
  • [10] H. A. Helfgott, Growth and generation in $\mathrm{SL}_{2}(\mathbb{Z}/p\mathbb{Z})$, Ann. of Math. (2) 167 (2008), 601–623.
  • [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.
  • [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.
  • [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.
  • [15] I. Ruzsa, Jr., On congruence preserving functions, Mat. Lapok. 22 (1971), 125–134.
  • [16] J. Sándor, D. S. Mitrinovic, and B. Crstici, Handbook of Number Theory, I, Springer, Dordrecht, 2006.
  • [17] T. Tao, Freiman’s theorem for solvable groups, Contrib. Discrete Math. 5 (2010), 137–184.
  • [18] T. Tao and V. Vu, Additive Combinatorics, Cambridge Stud. Adv. Math. 105, Cambridge Univ. Press, Cambridge, 2006.
  • [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.