bibkey: peltomaki2020abelianperiods
authors: J. Peltomäki
year: 2020
title: "Abelian periods of factors of Sturmian words"
doi: 10.1016/j.jnt.2020.04.007
url: https://arxiv.org/abs/1905.06138
claim: "Let $\\alpha = [0; \\overline{2}]$. The abelian period set of a Sturmian word of slope $\\alpha$ is $\\mathcal{Q}^+_\\alpha \\cup \\mathcal{M}_\\alpha$."
strata_touched:
  - D5/S1/Words/Mechanical/SilverSlopeAbelianPeriods
license: citation-only
triage: anchor

Abelian periods of factors of Sturmian words

J. Peltomäki, Abelian periods of factors of Sturmian words, Journal of Number Theory 214 (2020), 251–285, DOI 10.1016/j.jnt.2020.04.007, arXiv:1905.06138v3.

The paper’s Conjecture on page 283, line 1722, states verbatim:

Let $\alpha =[0;\overline{2}]$. The abelian period set of a Sturmian word of slope $\alpha$ is ${\mathcal{Q}}_{\alpha }^{+}\cup {\mathcal{M}}_{\alpha }$.

The paper defines the abelian period set as the set of minimum abelian periods of nonempty factors. It defines ${\mathcal{Q}}_{\alpha }^{+}$ as the denominators of convergents and semiconvergents and ${\mathcal{M}}_{\alpha }=\{t{q}_k:k\ge 0,1\le t\le a_{k+1}\}$. For the silver slope, the candidate set is $\{q_k\}\cup \{2q_k\}\cup \{q_k+q_{k-1}:k\ge 1\}$ with $q_0=1$, $q_1=2$, and $q_{k+1}=2q_k+q_{k-1}$.

The source states the conjecture after reporting computer experiments. The Lean theorem proves the equality: explicit Pell-phase constructions realise all three candidate families, and singular-window packing combined with approximation gaps excludes every noncandidate minimum period. The argument here is specific to the silver slope; the analogous question for constant partial quotients greater than two remains open.

Verified locator

DOI: https://doi.org/10.1016/j.jnt.2020.04.007

Source: https://arxiv.org/abs/1905.06138

Abelian definitions

The source states (arXiv v3, p. 7):

If $\mathcal{P}$ and $\mathcal{Q}$ are two Parikh vectors and $\mathcal{P}$ is componentwise less than or equal to $\mathcal{Q}$ but is not equal to $\mathcal{Q}$, then we say that $\mathcal{P}$ is contained in $\mathcal{Q}$.

Definition 2.4 states (arXiv v3, p. 8):

An abelian decomposition of a word $w$ is a factorization $w=u_0{u}_1\cdots u_{n-1}{u}_n$ such that $n\ge 2$, the words $u_1$, $\dots$, $u_{n-1}$ have a common Parikh vector $\mathcal{P}$ (i.e., they are abelian equivalent), and the Parikh vectors of $u_0$ and $u_n$ are contained in $\mathcal{P}$.

The common length $m$ of the words $u_1$, $\dots$, $u_{n-1}$ is called an abelian period of $w$. The minimum abelian period (i.e., the shortest) of $w$ is denoted by ${\mu }_w$.

The encoding uses Boolean letters, natural block lengths, a nonempty block list, and proper coordinatewise containment for both ends.