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

This paper is motivated by the following question in sieve theory. Given a subset $X\subset [N]$ and $α\in (0,1/2)$. Suppose that $|X\pmod p|\leq (α+o(1))p$ for every prime $p$. How large can $X$ be? On the one hand, we have the bound $|X|\ll_αN^α$ 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 $|X|\ll_αN^{O(α^{2014})}$ for small $α$). The result follows from studying the average size of $|X\pmod p|$ as $p$ varies, when $X=f(\mathbb{Z})\cap [N]$ is the value set of a polynomial $f(x)\in\mathbb{Z}[x]$.

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