On the Ordering and Injectivity of Lagrange Constants Associated with Certain Substitutions

@misc{yasutomi2026ordering,
  title = {{On the Ordering and Injectivity of Lagrange Constants Associated with Certain Substitutions}},
  author = {Shin-ichi Yasutomi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.32450},
  eprint = {2609.32450},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32450},
}

TY  - UNPB
TI  - On the Ordering and Injectivity of Lagrange Constants Associated with Certain Substitutions
AU  - Shin-ichi Yasutomi
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.32450
UR  - https://arxiv.org/abs/2609.32450
AB  - Motivated by the modern characterization of the Markoff spectrum via mechanical words and the Markoff Uniqueness Conjecture, we study the Lagrange constants of continued fractions generated by a parameterized family of substitutions ϕ_a,b defined by 0 ↦ aa and 1 ↦ bb with b = a + 1. Based on a three-fold arithmetic classification of pairs of rational numbers 0 ≤ x < y ≤ 1, we investigate the order relations between the corresponding Lagrange constants ℒ([ϕ_a,a+1(G(x))]) associated with the mechanical words G(x). While an inequality is established for pairs of the second type (type (2)) whenever a ≥ 1, the comparisons for type (1) and type (3) pairs hold for a ≥ 2. Consequently, for any integer a ≥ 2, where the order relations across all three types are completely determined, we establish an analogue of the Markoff Uniqueness Conjecture: the map x ↦ ℒ([ϕ_a,a+1(G(x))]) is injective on ℚ ∩ [0, 1]. Furthermore, supported by extensive numerical computations, we propose a general conjecture on the order relations for arbitrary b > a, highlighting a sharp phase transition in ordering behavior at the boundary b = a^2 + 2.
ER  -