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?

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    $\begingroup$ Thanks for the answers! I see it was a much simpler question than I thought. $\endgroup$ – Marcel K. Goh Nov 6 '20 at 0:07

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, $A=\bigcup_{n\in\mathbb{N}}\{n^3,n^3+1,\dots,n^3+n\}$.


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