OPEN
Let $A\subseteq \mathbb{N}$ be an infinite arithmetic progression and $f:A\to \{-1,1\}$ be a non-constant function. Must there exist a finite non-empty $S\subset A$ such that
\[\sum_{n\in S}\frac{f(n)}{n}=0?\]
What about if $A$ is an arbitrary set of positive density? What if $A$ is the set of squares excluding $1$?
Erdős and Straus
[ErSt75] proved this when $A=\mathbb{N}$. Sattler
[Sa75] proved this when $A$ is the set of odd numbers. For the squares $1$ must be excluded or the result is trivially false, since
\[\sum_{k\geq 2}\frac{1}{k^2}<1.\]