The inverse sieve conjecture

Let $X\subset [N]$ be a subset and let $\epsilon>0$ be real. Assume that for every real parameter $Q\geq N^{\epsilon}$,

$$\sum_{p\leq Q}\frac{|X\pmod p|}{p}\leq (1-\epsilon)\pi(Q).$$

Inverse sieve conjecture. At least one of the following holds: $|X|\ll_{\epsilon}N^{\epsilon}$, or there exists a polynomial $f(x)\in\mathbb{Q}[x]$ of degree $d\in[2,C]$ and height at most $N^C$ such that $|X\cap f([N])|\geq C^{-1}|X|$, where $C=C(\epsilon)$ is a constant. This conjecture predicts that a set occupying substantially fewer than the available residue classes on average is either very small or has a substantial intersection with a bounded-complexity polynomial value set; it is used in the paper to obtain an improved larger-sieve bound.