TITLE: Converse to Erdős' conjecture on arithmetic progressions
QUESTION [10 upvotes]: 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?

REPLY [17 votes]: 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.]

REPLY [16 votes]: It is not true. Take, for example, $A=\bigcup_{n\in\mathbb{N}}\{n^3,n^3+1,\dots,n^3+n\}$.