Erdős Problem 122

### What is the conjecture

https://www.erdosproblems.com/122

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
$$\frac\{\\#\\{ n\in \mathbb{N} : n+f(n)\in (x,x+F(x))\\}}{F(x)}\to \infty?$$

Status: open

### Prerequisites needed
I guess one has to decide on the set of "number theoretic functions" one want to formulate this conjectures, perhaps starting with those already in mathlib.

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else