OPEN

Let $n_1<n_2<\cdots$ be a sequence of integers such that
\[\limsup \frac{n_k}{k}=\infty.\]
Is
\[\sum_{k=1}^\infty \frac{1}{2^{n_k}}\]
transcendental?

Erdős [Er75c] proved the answer is yes under the stronger condition that $\limsup n_k/k^t=\infty$ for all $t\geq 1$.