Converse to Erdős' conjecture on arithmetic progressions

Unfortunately such a simple converse cannot be possible because one can "plant" long arithmetic progressions in $A$ while keeping it sparse overall. For example, let $A$ consist of all integers of the form $10^{n!}+m$ with $1 \leq m \leq n$ (which even makes $\sum_{n\in A} 1 / \log n$ converge).

[I see that GH from MO posted a very similar answer while I was editing mine.]

It is not true. Take, for example, $A=\bigcup_{n\in\mathbb{N}}\{n^3,n^3+1,\dots,n^3+n\}$.