# Converse to Erdős' conjecture on arithmetic progressions

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?

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

## 2 Answers

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