I apologise in advance if this has been asked here before. I did a search and did not find anything obvious. Erdős' conjecture states that if $A\subseteq {\bf N}$ is such that $\sum_{n\in A} n^{-1}$ diverges, then $A$ contains arbitrarily long arithmetic sequences.
I was wondering if anything is known about the converse statement; i.e., if $\sum_{n\in A} n^{-1}$ is finite, is it true that $A$ will not have $k$-term arithmetic progressions for $k$ large enough?
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, $A=\bigcup_{n\in\mathbb{N}}\{n^3,n^3+1,\dots,n^3+n\}$.
