Assume that $A\subset\mathbb{N}$ contains no 3-term arithmetic progression. It was conjectured by Erdős and Turán that $\sum_{a\in A}\frac{1}{a}$ converges. As far as I know this is open, although we are getting closer to a proof, see [Sanders' paper][1].


  [1]: http://front.math.ucdavis.edu/1011.0104