# The fractal dimension of the Kolakoski sequence is $2-1/e$

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:

• That sounds curious, but is there any indication it has any practical implications? – mathreadler Oct 28 '17 at 17:00
• @mathreadler Does it need to have a practical implication? Isn't curiosity enough? – Xander Henderson Oct 28 '17 at 20:00
• @XanderHenderson : No it doesn't need to. But I was curious. – mathreadler Oct 28 '17 at 20:01