Is $0.248163264128…$ a transcendental number?

Question

My question is in the title:

Is $a=0.248163264128…$ a transcendental number? The number $a$ is defined by concatenating the powers of $2$ (in base $10$).

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It is possible to express $a$ as a series :

$$a = \sum\limits_{n=1}^{\infty} 2^n \cdot 10^{ -\sum\limits_{k=1}^{n} (\lfloor{ k \cdot \log_{\,10}\,(2) }\rfloor + 1) } \tag{*}$$

I know that $a$ is irrational.

I know that if I consider the powers of $10$ instead the powers of $2$, i.e. if I consider $b=0.10100100010000...$, this number is transcendental.

Looking at the series (*), it seems very difficult to establish the transcendence of $a$. However, it is known (thanks to Kurt Mahler) that numbers as:

$$c = 0.149162536… = \sum\limits_{n=1}^{\infty} n^2 \cdot 10^{ -\sum\limits_{k=1}^{n} (\lfloor{ 2 \cdot \log_{\,10}\,(k) }\rfloor + 1) } \tag{**} $$

are transcendental ($c$ is the concatenation of the square numbers in base $10$ ; the same holds for third powers and so on).

I am aware that this could be a difficult problem. Similar numbers, as Copeland-Erdős constant, are not known to be transcendental. I would really appreciate if anyone had a reference about this number $a$, because I didn't find anything that could help me to determine whether $a$ is transcendental, or whether it is still unknown.

Thank you very much!

Answer 1

Yann Bugeaud "Distribution Modulo One and Diophantine Approximation", page 221:

For integers $b \geq 2$ and $c \geq 2$, let $(c)_b$ denote the sequence of digits of $c$ in its representation in base $b$. Mahler [471] proved that the real number $0 (c)_{10}(c^2)_{10} \dots$ is irrational. This was subsequently reproved and extended to every base $b \geq 2$ by Bundschuh [170] and Niederreiter [539]; see also [69, 172, 647, 652].

Problem 10.48. With the above notation, prove that, for arbitrary integers $b \geq 2$ and $c \geq 2$, Mahler's number $0 (c)_{b}(c^2)_{b} \dots$ is transcendental and normal to base $b$.

The question of normality of $0.248163264 \dots$ to base $10$ was already posed by Pillai [561].

Some of the links I was able to recover:

[172] P.Bundschuh, P. J.-S. Shiue and X.Y. Yu, Transcendence and algebraic independence connected with Mahler type numbers, Publ. Math. Debrecen 56 (2000), 121-130.

[647] Z.Shan, A note on irrationality of some numbers, J. Number Theory 25 (1987), 211-212.

[652] Irrationality criteria for numbers of Mahler's type. In: Analytic Number Theory (Kyoto, 1996), 343-351, London Math. Soc. Lecture Note Ser., 247, Cambridge University Press, Cambridge, 1997.

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Some data on this number from me. More digits:

$$0.2481632641282565121024204840968192163843276865536\dots$$

Simple continued fraction:

$$[0; 4, 33, 1, 3, 2, 565, 3, 5, 1, 10, 1, 43, 1, 1, 1, 1, 3, 1, 4, 1, 1, 3, 2, 3, 3, 2, 1, 1, 3, 5, 1, 16, 1, 15, 1, 2, 1, 3, 1, 3, 3, 327, \dots]$$

Euler type continued fraction:

$$\cfrac{1}{5-\cfrac{5}{6-\cfrac{5}{6-\cfrac{5}{51-\cfrac{50}{51-\cfrac{50}{51-\cfrac{50}{501-...}}}}}}}$$

The probability of a bigger partial quotent to occur after a smaller one in this fraction is equal to:

$$\frac{\ln 2}{\ln 10}=0.30103 \dots$$

Note that this fraction always approaches the number from below, fot example this truncation is exactly equal to $0.248163264128$

Unfortunately, general continued fractions do not afford any insight in the trancendentality of a number, as far as I know.