Helfgott–Venkatesh and Croot–Elsholtz inverse sieve conjecture

Let $S\subseteq \{0,\ldots,N\}$ satisfy $|S|\geq N^{\varepsilon}$ and occupy fewer than $\alpha p$ residue classes modulo every prime $p$, where $0<\alpha<1$. Helfgott–Venkatesh and Croot–Elsholtz inverse sieve conjecture. All but $O(N^{o(1)})$ elements of $S$ are contained in the set of values of a polynomial $f\in\mathbb{Z}[X]$ whose coefficients and degree are bounded in terms of $\alpha$ and $\varepsilon$. This conjecture seeks an inverse classification of sets that are badly distributed modulo primes: sufficiently large sets occupying few residue classes should have strong algebraic structure, generalizing the motivating examples given by quadratic-polynomial images. Its status is not established by the supplied text.