The Kolakoski sequence, which is defined as the
infinite sequence of symbols {1,2} that is its own run-length encoding (Wikipedia),
has been suggested to be self-similar$^{1}$. The fractral dimension of a self-similar time-series is directly related to its Hurst exponent$^{2}$. I have estimated the Hurst exponent of the Kolakoski sequence for different sequence lengths, and found that the answer converges near $1 / e$ when the sequence length increases. This suggests that
the fractral dimension of the Kolakoski sequence is $D=2-H = 2-1/e.$
Here is a small subset of the data
sequence length hurst exponent
1e2 .1167
1e3 .1796
1e4 .2236
1e5 .2579
1e6 .3108
1e7 .3464
1e8 .3657
2e8 .3720
3e8 .3766
My question is: Can you find a proof for this conjecture? If yes, I would be interested to write a brief note about this to a mathematical journal.
References:
$^{1}$ https://maths-people.anu.edu.au/~brent/pd/Kolakoski-ACCMCC.pdf
$^{2}$ https://en.wikipedia.org/wiki/Hurst_exponent#Relation_to_Fractal_Dimension