When is the sum of reciprocals of positive integers convergent?

Question

I'm looking for sufficient conditions on an infinite $\Lambda\subseteq\mathbb{Z}_+$ so that $$\sum_{n\in\Lambda}\frac{1}{n}<\infty.$$ I know that the contraposition of this question is given by Erdos' conjecture on arithmetic progressions, which says if $\sum_{n\in\Lambda}\frac{1}{n}=\infty$ then $\Lambda$ contains arbitrarily long arithmetic progressions, and there is decent computational evidence for this. Are there conjectures or (preferably) theorems for the contraposition? There are some obvious sufficient conditions, like $\Lambda$ can be indexed by a function growing like $n^p$ for some $p>1,$ but I'm looking for a weaker condition.