Examples of real-time transcendental number and superlinear-time trancsendental number

Question

Computation model is defined as Hartmanis and Stearns 4, it is well known that Liouvilles constant $$C_L=\sum_{i=1}^{\infty} 10^{-i!}$$ is computable in real time or linear time 1, 5 especially Theorem 12 in 1.

Is there any example of transcendental number computational complexity of which is $\Theta(n \log n)$, that is, any first $n$ digits of the 10-base expansion of it can be outputed by Turing Machine (defined by Hartmanis, Stearns, Yamada, Robin) in $\Theta(n \log n)$?

Please see the following reference for real-time computation or linear time computation if there is any ambiguity: 1, 2, 3, 4 5

Hope concrete example, if one want to discuss computation model, please show the computing code by the model. If one think the theorem 12 in 1 is not correct, please refute it in hard code.

Answer 1

Too long for a comment, though not a complete answer, sorry.

I don't understand the exact influence of allowing multiple tapes on complexity for Turing machines, but let me offer the following anyway.

Any number whose binary expansion is a so-called Sturmian or Beatty sequence (definition below) is transcendental; see "Transcendence of Numbers with a Low Complexity Expansion" by Ferenczi and Mauduit.

A Sturmian sequence $s$ is defined via an irrational number $x$, and the nth bit is $s_n = \lfloor (n+1) x \rfloor - \lfloor nx \rfloor$. Put another way, the nth bit is 1 iff there is an integer between $nx$ and $(n+1)x$.

Here's where I guess I can't give an explicit example, but I would think that by using numbers with various computability rates (since $x$ can be any irrational at all!), I would guess that you could achieve any superlinear runtime, including $n \ln n$. Perhaps someone else can be a little more explicit here.