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?