Computation model is defined as Hartmanis and Stearns [4](https://www.ams.org/journals/tran/1965-117-00/S0002-9947-1965-0170805-7/S0002-9947-1965-0170805-7.pdf), it is well known that Liouvilles constant
$$C\_L=\sum\_{i=1}^{\infty} 10^{-i!}$$ is computable in real time or linear time [1](https://www.ams.org/journals/tran/1965-117-00/S0002-9947-1965-0170805-7/S0002-9947-1965-0170805-7.pdf), [5](https://rjlipton.wpcomstaging.com/2012/06/15/why-the-hartmanis-stearns-conjecture-is-still-open/) especially Theorem 12 in [1](https://www.ams.org/journals/tran/1965-117-00/S0002-9947-1965-0170805-7/S0002-9947-1965-0170805-7.pdf).


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](https://link.springer.com/article/10.1007/BF02759719),
[2](https://ieeexplore.ieee.org/document/5219459),
[3](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/hisao-yamada-realtime-computation-and-recursive-functions-not-realtime-computable-ire-transactions-on-electronic-computers-vol-ec11-1962-pp-753760/06282D96A6D1C5F481E9A8B2B5A55535),
[4](https://www.ams.org/journals/tran/1965-117-00/S0002-9947-1965-0170805-7/S0002-9947-1965-0170805-7.pdf)
[5](https://rjlipton.wpcomstaging.com/2012/06/15/why-the-hartmanis-stearns-conjecture-is-still-open/)


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](https://www.ams.org/journals/tran/1965-117-00/S0002-9947-1965-0170805-7/S0002-9947-1965-0170805-7.pdf) is not correct, please refute it in hard code.



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


