# On the irrationality measure of $\sum_{n=1}^\infty a^{-b^n}$

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.

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Somewhat related: Bailey and Borwein have been studying $\sum c^{-n}b^{-c^n}$. See, e.g., Bailey, David H.; Borwein, Jonathan M., Nonnormality of Stoneham constants, Ramanujan J. 29 (2012), no. 1-3, 409–422, MR2994109 . – Gerry Myerson Oct 21 '13 at 22:35
Thank you, Gerry. This and Vesselin's answer&comments below suggest to generalize the questions in the OP to the case of "generalized Stoneham constants". Really hoping this is OK with the policy of MO, I've just opened another thread to discuss about this: mathoverflow.net/questions/145554/…. – Salvo Tringali Oct 22 '13 at 17:15

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).

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Thanks, Vesselin, this completely answers Q1 (and much more). Anything about the other questions? – Salvo Tringali Oct 21 '13 at 22:16
Q2: $\xi_{a,b}$ is never a Liouville number, and in fact, Q3: its irrationality measure is $b$. The former holds in the much more general context of automatic numbers. Here is one reference: a recent paper by J. Bell, Y. Bugeaud, and M. Coons, arxiv.org/pdf/1307.4123.pdf . – Vesselin Dimitrov Oct 21 '13 at 22:32
Thanks a lot! So it's even known that the irrationality measure, say $\mu_{a,b}$, of $\xi_{a,b}$ is $b$. Indeed, I'd appreciate much a reference to this result: I had guessed it, but unhappily the best that I've been able to come up is nothing more than $\mu_{a,b} \ge b$. – Salvo Tringali Oct 21 '13 at 22:53
This is mentioned as a well-known fact on page 2 in a paper math.univ-lyon1.fr/~adamczew/IMAN.pdf by Adamczewski and Rivoal, but I don't know of an actual reference (sorry!). By the way, note that for $b \neq 2$, the answer to Q1 is immediate by Roth's theorem and your observation that $\mu_{a,b} \geq b$. – Vesselin Dimitrov Oct 21 '13 at 23:09
(...) any automatic number is Mahler in view of a theorem by P.-G. Becker, namely Theorem 1 in $k$-regular power series and Mahler-type functional equations, JNT 49(3): 269-286, 1994. – Salvo Tringali Oct 22 '13 at 13:28