I study infinite aperiodic sequences like Thue-Morse.

Simple substitution rules allow you to get even more complex.
I'm interested in the distribution of tuples in such sequences.
For example, in Thue-Morse $1$ and $0$ represent equivalently,
of the 2-tuples $\{00, 01,10,11\}$ represented all four, but not equally,
and of the eight 3-tuples present only six.

For a strictly chaotic binary sequence, all $n$-tuples must be represented uniformly,
each type with a probability of $2^{-n}$.
If you plot their distribution (sorted by descending, in Log-Log scale), it will be the following:

enter image description here

For sufficiently large $n$ we see that the plot is not horizontal,
this is obviously related to limited sample size (in this case it was only $50000$).
I’m interested in the real shape of these plots, especially the "tails" on the right,
to compare with quasi-random sequences.
For example, rule $11 \to 00, 00 \to 01, 0 \to 10$ generates a rich enough sequence with this distribution of tuples:

enter image description here

I’d like to know what is the effect of quasi-chaotic, and what is the limit of the sample.
To do this, I need to know the approximate shape (equation) for the tail of tuples distributions.



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