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

Shao, Xuancheng

doi:10.2140/ant.2015.9.2325

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

Xuancheng Shao

Vol. 9 (2015), No. 10, 2325–2346

DOI: 10.2140/ant.2015.9.2325

Abstract

This paper is motivated by the following question in sieve theory. Given a subset $X \subset [N]$ and $α \in (0, \frac{1}{2})$ . Suppose that $| X (mod p) | \leq (α + o (1)) p$ for every prime $p$ . How large can $X$ be? On the one hand, we have the bound $| X | ≪_{α} N^{α}$ 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 $| X | ≪_{α} N^{O (α^{2014})}$ for small $α$ ). The result follows from studying the average size of $| X (mod p) |$ as $p$ varies, when $X = f (ℤ) \cap [N]$ is the value set of a polynomial $f (x) \in ℤ [x]$ .

Keywords

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

Mathematical Subject Classification 2010

Primary: 11N35

Secondary: 11R45, 11R09

Milestones

Received: 17 December 2014

Revised: 19 July 2015

Accepted: 17 August 2015

Published: 16 December 2015

Authors

Xuancheng Shao
Mathematical Institute University of Oxford Radcliffe Observatory Quarter Woodstock Road Oxford OX2 6GG United Kingdom

Vol. 9, No. 10, 2015