# On the irrationality measure of generalized Stoneham numbers

Pick non-zero integers $a,b,c$ with $a,b \ge 2$ and let $\xi_{a,b,c}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n} c^{-n}$ (no restriction is made on the sign of $c$); when $b = c$ and $\gcd(a,b) = 1$, the sum is called a Stoneham number (whence the title of the thread). Stoneham numbers owe their glory to the fact of being $b$-normal (and certainly to other non-trivial properties of which I'm not aware), a result due to D. H. Bailey and R. E. Crandall (see Random generators and normal numbers, Experimental Mathematics, 11(4): 527-546, 2002). I learned about this after a comment by Gerry Myerson to the OP of an another thread, of which this one is just a follow-up (I hope this is fine: If I'm not too wrong, the policy here is to avoid too many questions in one thread or substantial edits of existing threads, especially when they've already received answers). Building on these premises, I have the following:

Q1. Is it known whether $\xi_{a,b,c}$ is a Mahler number for [any, some] $c \ne 1$?

Let me recall that a Mahler number is a number of the form $F(1/q)$, where

1. $F(x) \in \mathbb{Q}[[x]]$ satisfies a functional equation of the form $\sum_{i=0}^d a_i(x) F(x^{k^i})=0$ for integers $d\ge 1$ and $k \ge 2$ and polynomials $a_0(x),\ldots,a_d(x)\in\mathbb{Z}[x]$ with $a_0(x)a_d(x)\ne 0(x)$.
2. $q$ is an integer $\ge 2$ such that $1/q$ is in the circle of convergence of $F(x)$.

In addition to this (and partially conditioned to the answer to Q1), the next question seems quite natural:

Q2. What is known about the "diophantine nature" (rational vs irrational, algebraic vs transcendent) and/or the irrationality measure, say $\mu_{a,b,c}$, of $\xi_{a,b,c}$ in the case when $c \ne 1$?

As for Q2, the special case when $c = 1$ is completely answered by Vesselin in the other thread mentioned above, even though I'd really appreciate if anybody would provide a reference to a published proof of the fact that $\mu_{a,b,1} = b$ (as pointed out by the same Vesselin, this appears as a remark, with no proof or reference, on p. 2 of a paper by B. Adamczewski and T. Rivoal, namely Irrationality measures for some automatic real numbers, Math. Proc. Cambridge Phil. Soc. 147: 659-678, 2009).

Thanks in advance for any help.

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$\mu_{a,b,1} = b$ should follow from the description of the partial fraction expansion of $\xi_{a,b,1}$ in J. Shallit's "Simple continued fractions for some irrational numbers," J. Number Theory 11, 1979. (I haven't looked into that paper, though.) Alternatively, for $\mu_{a,b,b} \leq b$, I would suggest looking at the proof of Theorem 2.2 in Adamczewski and Cassaigne's Compositio paper ("Diophantine properties of real numbers generated by finite automata," 2006), or at the proof of Lemma 4 in L. Levesley, C. Salp, and S. Velani's "On a problem of K. Mahler..." –  Vesselin Dimitrov Oct 22 '13 at 17:55
Thanks, Vess, I will check all of this. And yes, there is a typo in the OP: $\mu_{a,b,b} = b$ should be $\mu_{a,b,1} = b$ (going to fix it). By the way, is there anything that I can do to merge this user profile with mathoverflow.net/users/41663/user41663? –  Salvo Tringali Oct 22 '13 at 18:23
A minor remark: $\xi_{a,b,b}$ is clearly irrational when $\gcd(a,b) = 1$ (as a consequence of Bailey and Crandall's extension of Stoneham's $b$-normality theorem), but is there any "obvious" reason why the same should continue to be true when $\gcd(a,b) \ge 2$? –  Salvo Tringali Oct 22 '13 at 23:44
Please request merger here. –  François G. Dorais Oct 23 '13 at 0:28