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?



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.]


