Vol. 9, No. 10, 2015

Download this article
Download this article For screen
For printing
Recent Issues

Volume 16
Issue 2, 231–519
Issue 1, 1–230

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
Other MSP Journals
Polynomial values modulo primes on average and sharpness of the larger sieve

Xuancheng Shao

Vol. 9 (2015), No. 10, 2325–2346

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].

Gallagher's larger sieve, inverse sieve conjecture, value sets of polynomials over finite fields
Mathematical Subject Classification 2010
Primary: 11N35
Secondary: 11R45, 11R09
Received: 17 December 2014
Revised: 19 July 2015
Accepted: 17 August 2015
Published: 16 December 2015
Xuancheng Shao
Mathematical Institute
University of Oxford
Radcliffe Observatory Quarter
Woodstock Road
United Kingdom