Steward erdos-257

Let $A\subseteq\mathbb{N}$ be an infinite set. Is $$ \sum_{n\in A} \frac{1}{2^n - 1} $$ irrational?

Its steward: Michael Higgins self-admitted. A problem may have several, who hold the role jointly.

This problem has a steward, so a new steward joins by invitation: a member of its panel invites you by your GitHub login and you accept in the problem’s Panel section, or a curator adds you directly (D-32). Co-stewards are welcome: to ask, write to a steward or open an issue. The panel’s rules →

What the role asks

Understand what the network proves on this problem.
Write it up, and sign the explainer of the proof that closes it.
Best efforts, no deadline: step down at any time.