OPEN
Let $Y(x)$ be the minimal $y$ such that there exists a choice of congruence classes $a_p$ for all primes $p\leq x$ such that every integer in $[1,y]$ is congruent to at least one of the $a_p\pmod{p}$.
Give good estimates for $Y(x)$. In particular, can one prove that $Y(x)=o(x^2)$ or even $Y(x)\ll x^{1+o(1)}$?
This function is closely related to the problem of gaps between primes (see
[4]). The best known upper bound is due to Iwaniec
[Iw78],
\[Y(x) \ll x^2.\]
The best lower bound is due to Ford, Green, Konyagin, Maynard, and Tao
[FGKMT18],
\[Y(x) \gg x\frac{\log x\log\log\log x}{\log\log x},\]
improving on a previous bound of Rankin
[Ra38].
Maier and Pomerance have conjectured that $Y(x)\ll x(\log x)^{2+o(1)}$.
See also [688] and [689].
