Let set $C$: $C \subset \mathbb{Z}^+$, and give $c \in \mathbb{Z}^+, c > 1$. if $\sum_{k \in C}\frac{1}{c^k}$ is an algebraic number, for other $p \in \mathbb{Z}^+, p > 1$, is the number $\sum_{k \in C}\frac{1}{p^k}$ an algebraic number too?
If $C$ is a finite set, it is obvious. And infinite set?
If $x=\sum_{k\in C} 1/c^k$ for some integer $c\ge 2$, then the base-$c$ expansion of $x$ has digits $1$ for $k \in C$ and $0$ otherwise.
If $x$ is rational, the base-$c$ expansion of a rational number is eventually periodic, and this implies that the sum is rational for all $p$.
It is conjectured that every irrational algebraic number is normal in every base. This would imply that your sum can't be an irrational algebraic for $c > 2$. But we are very far from being able to prove that conjecture.