Problem 122

Erdős Problem #122: declared status 'open'. Formalized: no. For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $f(n)/F(n)\to 0$ for almost all $n$, there are infinitely many $x$ such that$$\frac{\#\{ n\in \mathbb{N} : n+f(n)\in (x,x+F(x))\}}{F(x)}\to \infty?$$ Prize: no. Tags: number theory.