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

Abstract

This paper is motivated by the following question in sieve theory. Given a subset $X\subset [N]$ and $\alpha \in (0,1/2)$. Suppose that $|X(\mathrm{mod}p)|\le (\alpha +o(1))p$ 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 $|X|{\ll }_{\alpha }{N}^{O({\alpha }^{2014})}$ for small $\alpha$). The result follows from studying the average size of $|X(\mathrm{mod}p)|$ as $p$ varies, when $X=f(\mathbb{Z})\cap [N]$ is the value set of a polynomial $f(x)\in \mathbb{Z}[x]$.

18 pages

References in corpus (2)

• Hilbert's irreducibility theorem and the larger sieve
• On an inverse ternary Goldbach problem