#### 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 $X\subset \left[N\right]$ and $\alpha \in \left(0,\frac{1}{2}\right)$. Suppose that $|X\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}p\right)|\le \left(\alpha +o\left(1\right)\right)p$ for every prime $p$. How large can $X$ be? On the one hand, we have the bound $|X|{\ll }_{\alpha }{N}^{\alpha }$ 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 $|X|{\ll }_{\alpha }{N}^{O\left({\alpha }^{2014}\right)}$ for small $\alpha$). The result follows from studying the average size of $|X\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}p\right)|$ as $p$ varies, when $X=f\left(ℤ\right)\cap \left[N\right]$ is the value set of a polynomial $f\left(x\right)\in ℤ\left[x\right]$.

##### 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 