Suppose $X\subset \Bbb Z_{<0}$, and $a=\sum_{n\in X} 2^n$ is algebraic, then is $b=\sum_{n\in X}3^n$ also algebraic? (To clarify, what I mean is: does it hold for all such $X$?)
We know that $a$ and $b$ must either both be rational or both irrational, simply by noting the fact that under any integer base, rational numbers coincide exactly with recurring decimals. But what about being algebraic/transcendental? Is there anything we can say about it? By the way I'm no algebraist (not even a smart math student) so apologies for not being able to proceed any deeper.