trureturing — Discovering truth

bibkey: dekking2022twoblock authors: F. M. Dekking and M. Keane year: 2022 title: Two-block substitutions and morphic words doi: 10.48550/arXiv.2202.13548 url: https://arxiv.org/abs/2202.13548v1 claim: Conjecture 4 of v1 asserts that the frequency of 1 in the Thue-Morse two-block fixed point with prefix 00 exists and equals one half. strata_touched:

• D5/S1/Words/TwoBlockSubstitution/ThueMorseFrequency license: citation-only triage: anchor

Verified locator

DOI: 10.48550/arXiv.2202.13548

Source: https://arxiv.org/abs/2202.13548v1

The source word

Section 4, “Thue-Morse meets Kolakoski”, p. 5 of v1:

We consider the two-block substitution ${\kappa }_{\mathrm{TM}}$ defined by ${\kappa }_{\mathrm{TM}}(00)=001,{\kappa }_{\mathrm{TM}}(01)=010,{\kappa }_{\mathrm{TM}}(10)=101,{\kappa }_{\mathrm{TM}}(11)=110.$

The fixed point $x=x^{(00)}=x_0{x}_1\dots$ of the two-block morphism ${\kappa }_{\mathrm{TM}}$ with prefix 00 satisfies very similar recurrence relations: $x_{3n}=x_{2n},x_{3n+1}=x_{2n+1},x_{3n+2}=1-x_{2n+1}.$

The encoding uses false for 0, true for 1 and natural indices starting at zero. The recursion reads the residue before making a recursive call; its prefix is 00. The source prints $x^{(00)}=001110101101110010110001101110001\dots$.

Conjecture 4

Section 4, p. 6 of v1:

Conjecture 4 The frequency of 1 in $x^{(00)}$ exists and equals $\frac{1}{2}$.

Frequency means the limit of the number of indices $n<N$ with $x_n=1$, divided by $N$. The formal statement uses a real-valued Filter.Tendsto of this ratio along Filter.atTop. The value at $N=0$ is immaterial to the limit.

This locator is specifically v1. ArXiv v2 and the journal version, Advances in Applied Mathematics 148 (2023), 102536, change the substitution. The present statement concerns the table quoted above.

Scope of the conclusion

The coefficient functional of the signed substitution has zero adjacent cyclic correlation and squared energy $3^{k-1}$ for $k\ge 1$. Finite Cauchy–Schwarz bounds aligned block sums; a prefix decomposition makes their normalized sum tend to zero. This proves the quoted single-letter frequency assertion. Mirror invariance, quadratic complexity, uniform recurrence and frequencies of longer factors are separate questions.