A uniformness conjecture of the Kolakoski sequence, graph connectivity, and correlations

TL;DR

This paper investigates the Kolakoski sequence, proposing a generalized uniformness conjecture, and analyzes autocorrelation patterns using graph connectivity and periodic sequences, revealing complex wave-like behaviors.

Contribution

It introduces the generalized uniformness conjecture, proves related graph connectivity results, and develops a method to compute autocorrelations for large distances in the sequence.

Findings

Autocorrelation functions exhibit wave-like patterns with common nodes.

Graph connectivity of certain directed graphs is established unconditionally.

Correlation estimates suggest complex, residue-dependent behaviors in the sequence.

Abstract

The Kolakoski sequence is the unique infinite sequence with values in $\{1,2\}$ and first term $1$ which equals the sequence of run-lengths of itself, we call this $K(1,2).$ We define $K(m,n)$ similarly. A well-known conjecture is that the limiting density of $K(1,2)$ is one-half. We state a natural generalization, the "generalized uniformness conjecture" (GUC). The GUC seems intractable, but we prove a partial result. The GUC implies that members of a certain family of directed graphs $G_{m,n,k}$ are all strongly connected. We prove this unconditionally. For $d>0,$ let $cf(m,n,d)$ be the density of indices $i$ such that $K(m, n)_i=K(m, n)_{i+d}.$ Essentially, $cf(m, n, d)$ is the autocorrelation function of a stationary stochastic process with random variables $\{X_t\}_{t\in\mathbb{Z}}$ whereby a sample of a finite window of this process is formed by copying as many consecutive terms of $K(m,n)$ starting from a "uniformly random index" $i\in\mathbb{Z}_+.$ Assuming the GUC, we prove that we can compute $cf(m,n,d)$ exactly for quite large $d$ by constructing a periodic sequence $S$ of period around $10^{8.5}$ such that for $d$ not too large, the correlation frequency at distance $d$ in $K(m,n)$ equals that in $S.$ We efficiently compute correlations in $S$ using polynomial multiplication via FFT. We plot our estimates $cf(m,n,d)$ for several small values of $(m,n)$ and $d\le10^5$ or $10^6$. We note many suggested patterns. For example, for the three pairs $(m,n)\in\{(1,2),(2,3),(3,4)\},$ the function $cf(m,n,d)$ behaves very differently as we restrict $d$ to the $m+n$ residue classes $\text{mod}$ $m+n.$ The plots of the three functions $cf(1,2,d),cf(2,3,d),$ and $cf(3,4,d)$ resemble waves which have common nodes. We consider this very unusual behavior for an autocorrelation function. The pairs $(m,n)\in\{(1,4),(1,6),(2,5)\}$ show wave-like patterns with much more noise.