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