# Is 0.1010010001000010000010000001…0.1010010001000010000010000001 \ldots transcendental?

Does anyone know if this number is algebraic or transcendental, and why?

The number $0.1010010001000010000010000001\ldots$ is transcendental.

Consider following three Jacobi theta series defined by

and for any $m \in \mathbb{Z}_{+}$, $k \in \{ 2, 3, 4 \}$, use
$\displaystyle D^m\theta_k(q)$ as a shorthand for
$\displaystyle \left( q\frac{d}{dq} \right)^m \theta_k(q)$.

Based on Corollary 52 of a survey article Elliptic functions and Transcendence by M. Waldschmidt in 2006,

Let $i, j$ and $k \in \{ 2,3,4 \}$ with $i \ne j$. Let $q \in \mathbb{C}$
satisfy $0 < |q| < 1$. Then each of the two fields

has transcendence degree $\ge 3$ over $\mathbb{Q}$

We know for any non-zero algebraic $q$ with $|q| < 1$, the three $\theta_k(q)$, in particular $\theta_2(q)$ is transcendental. Since

and using the fact $\frac{1}{\sqrt{10}}$ and $\frac{\sqrt[8]{10}}{2}$ are both algebraic, we find the number at hand is transcendental.