Vol. 9, No. 10, 2015

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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
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(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].

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