Erdős problem 26

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 $a\in A$? The question as posed follows negatively from Davenport–Erdős (1951). The AI result settles Tenenbaum's harder variant, also negatively: there is an infinite $A$ such that for every $k\geq 1$ the set of multiples of $A+k$ has upper density below $0.34$.