Algebra & Number Theory

Polynomial values modulo primes on average and sharpness of the larger sieve

Xuancheng Shao

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This paper is motivated by the following question in sieve theory. Given a subset X [N] and α (0, 1 2). Suppose that |X(modp)| (α + o(1))p for every prime p. How large can X be? On the one hand, we have the bound |X|αNα from Gallagher’s larger sieve. On the other hand, we prove, assuming the truth of an inverse sieve conjecture, that the bound above can be improved (for example, to |X|αNO(α2014) for small α). The result follows from studying the average size of |X(modp)| as p varies, when X = f() [N] is the value set of a polynomial f(x) [x].

Article information

Algebra Number Theory, Volume 9, Number 10 (2015), 2325-2346.

Received: 17 December 2014
Revised: 19 July 2015
Accepted: 17 August 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11N35: Sieves
Secondary: 11R45: Density theorems 11R09: Polynomials (irreducibility, etc.)

Gallagher's larger sieve inverse sieve conjecture value sets of polynomials over finite fields


Shao, Xuancheng. Polynomial values modulo primes on average and sharpness of the larger sieve. Algebra Number Theory 9 (2015), no. 10, 2325--2346. doi:10.2140/ant.2015.9.2325.

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  • J. C. Andrade, L. Bary-Soroker, and Z. Rudnick, “Shifted convolution and the Titchmarsh divisor problem over $\mathbb{F}\sb q[t]$”, Philos. Trans. A 373:2040 (2015), 20140308, 18.
  • E. Bank, L. Bary-Soroker, and L. Rosenzweig, “Prime polynomials in short intervals and in arithmetic progressions”, Duke Math. J. 164:2 (2015), 277–295.
  • B. J. Birch and H. P. F. Swinnerton-Dyer, “Note on a problem of Chowla”, Acta Arith. 5 (1959), 417–423.
  • E. Bombieri and J. Pila, “The number of integral points on arcs and ovals”, Duke Math. J. 59:2 (1989), 337–357.
  • S. D. Cohen, “The distribution of polynomials over finite fields”, Acta Arith. 17 (1970), 255–271.
  • E. S. Croot, III and C. Elsholtz, “On variants of the larger sieve”, Acta Math. Hungar. 103:3 (2004), 243–254.
  • E. S. Croot, III and V. F. Lev, “Open problems in additive combinatorics”, pp. 207–233 in Additive combinatorics, CRM Proc. Lecture Notes 43, Amer. Math. Soc., Providence, RI, 2007.
  • R. Dietmann, “Probabilistic Galois theory”, Bull. Lond. Math. Soc. 45:3 (2013), 453–462.
  • J. S. Ellenberg, C. Elsholtz, C. Hall, and E. Kowalski, “Non-simple abelian varieties in a family: geometric and analytic approaches”, J. Lond. Math. Soc. $(2)$ 80:1 (2009), 135–154.
  • C. Elsholtz and A. J. Harper, “Additive decompositions of sets with restricted prime factors”, Trans. Amer. Math. Soc. 367:10 (2015), 7403–7427.
  • A. Entin, “On the Bateman–Horn conjecture for polynomials over large finite fields”, preprint, 2014.
  • M. Fried, “On a conjecture of Schur”, Michigan Math. J. 17 (1970), 41–55.
  • P. X. Gallagher, “A larger sieve”, Acta Arith. 18 (1971), 77–81.
  • P. X. Gallagher, “The large sieve and probabilistic Galois theory”, pp. 91–101 in Analytic number theory (St. Louis, MO, 1972), Proc. Sympos. Pure Math. 24, Amer. Math. Soc., Providence, R.I., 1973.
  • J. Gomez-Calderon and D. J. Madden, “Polynomials with small value set over finite fields”, J. Number Theory 28:2 (1988), 167–188.
  • B. Green and A. J. Harper, “Inverse questions for the large sieve”, Geom. Funct. Anal. 24:4 (2014), 1167–1203.
  • R. Guralnick and D. Wan, “Bounds for fixed point free elements in a transitive group and applications to curves over finite fields”, Israel J. Math. 101 (1997), 255–287.
  • D. R. Heath-Brown, “The density of rational points on curves and surfaces”, Ann. of Math. $(2)$ 155:2 (2002), 553–595.
  • H. A. Helfgott and A. Venkatesh, “How small must ill-distributed sets be?”, pp. 224–234 in Analytic number theory, Cambridge Univ. Press, 2009.
  • M. Hindry and J. H. Silverman, Diophantine geometry, Graduate Texts in Mathematics 201, Springer, New York, 2000.
  • E. Kowalski, “Some aspects and applications of the Riemann hypothesis over finite fields”, Milan J. Math. 78:1 (2010), 179–220.
  • J. C. Lagarias and A. M. Odlyzko, “Effective versions of the Chebotarev density theorem”, pp. 409–464 in Algebraic number fields: $L$-functions and Galois properties (Durham, UK, 1975), edited by A. Fr öhlich, Academic Press, London, 1977.
  • S. Lang and A. Weil, “Number of points of varieties in finite fields”, Amer. J. Math. 76 (1954), 819–827.
  • H. L. Montgomery, “The analytic principle of the large sieve”, Bull. Amer. Math. Soc. 84:4 (1978), 547–567.
  • W. M. Schmidt, Equations over finite fields: an elementary approach, Lecture Notes in Mathematics 536, Springer, Berlin, 1976.
  • J.-P. Serre, Topics in Galois theory, 2nd ed., Research Notes in Mathematics 1, A K Peters, Wellesley, MA, 2008.
  • X. Shao, “On an inverse ternary Goldbach problem”, preprint, 2014. To appear in Amer. J. Math.
  • M. N. Walsh, “The inverse sieve problem in high dimensions”, Duke Math. J. 161:10 (2012), 2001–2022.
  • M. N. Walsh, “Bounded rational points on curves”, Int. Math. Res. Not. 2015:14 (2015), 5644–5658.
  • D. Zywina, “Hilbert's irreducibility theorem and the larger sieve”, preprint, 2010.