Yes, I am. This is common in additive theory (e.g., see Ruzsa's survey Sumsets and structure).

After a closer look, Theorem 1 in Becker's paper proves something stronger, namely: Any regular number (in the sense of Allouche and Shallit) is Mahler.

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 $\gcd(a,b) \ge 2$?

Well, I beg to strongly disagree with the statement that differential equations are "an indispensable tool in proving transcendence".

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?

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/….

(...) 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.

A minor follow-up to Vesselin's first comment: In the paper by J. Bell, Y. Bugeaud and M. Coons, it is in fact proved that any Mahler number is not Liouville. Here, a Mahler number is a number of the form $F(1/b)$, where (i) $F(x)\in\mathbb{Q}[[x]]$ satisfies a functional equation of the form $\sum_{i=0}^da_i(x)F(x^{k^i})=0$ for integers $d \ge 1$ and $k \ge 2$ and polynomials $a_0(x),\ldots,a_d(x)\in\mathbb{Z}[x]$; (ii) $1/b$ is in the circle of convergence of $F(x)$; and (iii) $a_0(x)a_d(x)\ne 0(x)$. This is more general than saying that automatic numbers are not Liouville because (...)

Thanks again! Yes, Roth's theorem is exactly what I had in mind for the proof that $\xi_{a,b}$ is transcendent for $b > 2$.

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 $\mu_{a,b} \ge b$.

Thanks, Vesselin, this completely answers Q1 (and much more). Anything about the other questions?