SOLVED

Let $A\subset\mathbb{N}$ be infinite. Must there exist some $k\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\in A$?

Asked by Erdős and Tenenbaum. Ruzsa gave the following simple counterexample: let $A=\{n_1<n_2<\cdots \}$ where $n_l \equiv -(k-1)\pmod{p_k}$ for all $k\leq l$, where $p_k$ denotes the $k$th prime.

Tenenbaum asked the weaker variant (still open) where for every $\epsilon>0$ there is some $k=k(\epsilon)$ such that at least $1-\epsilon$ density of all integers have a divisor of the form $a+k$ for some $a\in A$.