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](http://www.math.sc.edu/~filaseta/gradcourses/Math785/Math785Notes5.pdf) (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:



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> > **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 $\xi\_{a,b}$?
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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](http://en.wikipedia.org/wiki/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:



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> > **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]?
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Based on Q2, it is now natural to ask the following:



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> > **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 $> 2$?
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Thanks in advance for any help.



On the irrationality measure of $\sum_{n=1}^\infty a^{-b^n}$
This result is due originally to K. Mahler, and holds true more generally with any algebraic $a$ having $|a| > 1$ (so that the series converges absolutely). I can recommend Masser's lecture in the CIME 2000 school on diophantine approximations (LNM 1819), where the main idea of Mahler's proof is outlined as an illustration of the typical transcendence proof. The complete argument can be found in the opening chapter (Theorem 1.1.2) of K. Nishioka's book "Mahler Funtions and Transcendence" (LNM 1631), where you may also find various related results and generalizations.


Mahler's proof of the transcendence of $f(a^{-1})$ is based on the functional equation $f(z^b) = f(z) - z$ of the series $f(z) := \sum\_{n \geq 1} z^{b^n}$. A different approach, which is based on Schmidt's Subspace theorem and allows for much more general transcendence statements, was discovered by P. Corvaja and U. Zannier in their article "Some new applications of the subspace theorem" (*Compositio math*, 2002).


