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

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Abstract

This paper is motivated by the following question in sieve theory. Given a subset X ⊂ [N] and α ∈ (0,1/2). Suppose that |X (mod p)| ≤ (α + o(1)) p for every prime p. How large can X be? On the one hand, we have the bound |X| ≪α Nα 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|≪ αNO2014) for small α). The result follows from studying the average size of |X (mod p)| as p varies, when X = f (ℤ) ∩[N] is the value set of a polynomial f (x) ∈ ℤ[x].

Original languageEnglish
Pages (from-to)2325-2346
Number of pages22
JournalAlgebra and Number Theory
Volume9
Issue number10
DOIs
StatePublished - 2015

Bibliographical note

Publisher Copyright:
© 2015 Mathematical Sciences Publishers.

Keywords

  • Gallagher’s larger sieve
  • Inverse sieve conjecture
  • Value sets of polynomials over finite fields

ASJC Scopus subject areas

  • Algebra and Number Theory

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