Pick integers $a, b \ge 2$ and let $\xi_{a,b}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n}$. It is known that $\xi_{2,2}$ is transcendental: I learned a proof of this from notes by M. Filaseta (who attributes the basic idea to P. Erdős, but does not provide a reference), and I'm confident that the argument can be made to work at least in the more general case when $a = b$, but I have contrasting feelings about the case $a \ne b$ (and no much time to look closer at this right now). Then, my first question is:

Q1.Could you kindly provide me with a reference to Erdős' original work on the subject (if ever published) or any published work addressing the more general question of the transcendence of $\xi_{a,b}$?

While I've my own proof for the general result in the above, I've also read that $\xi_{2,2}$ is known not to be a Liouville number, and I guess that the same holds true for $\xi_{a,b}$ at least in the case $a = b$ (but I don't have a clue on how to prove it). So my second question is:

Q2.Is it known whether $\xi_{a,b}$ is never a Liouville number? If [yes, no], could you provide a reference to a published work where this is [proved, disproved]?

Based on Q2, it is now natural to ask the following:

Q3.What is known about the irrationality measure of $\xi_{a,b}$? In particular, is it known whether [each, any] of the $\xi_{a,b}$ has an irrationality measure $> 2$?

Thanks in advance for any help.