PreprintCC0arXiv
Transcendence criteria for the minimal word of the rational base 3/2
Abstract
Let x0 be a positive integer, let xn=⌈ 3xn−1/2⌉, and let wn=2xn+1−3xn∈{0,1} be the associated word studied by Dubickas; for x0=1 the orbit is A061419 and w is the minimal word g3/2 of the rational base number system of Akiyama, Frougny and Sakarovitch. The orbit encodes a real constant K=limnxn(2/3)n, equal for x0=1 to ω3/2=K(3)=1.6222705028…, whose irrationality has been open since 1977. We prove that if w is automatic then K is transcendental; equivalently, an algebraic K forces w to be non-automatic. Further we prove that either the complexity of w exceeds every linear bound or K is irrational. All results are formally verified in the Lean~4 proof assistant, on three cited axioms.
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BibTeX
@misc{stephan2026transcendence,
title = {{Transcendence criteria for the minimal word of the rational base $3/2$}},
author = {Ralf Stephan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19007},
eprint = {2609.19007},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19007},
note = {CC0},
}
RIS
TY - UNPB
TI - Transcendence criteria for the minimal word of the rational base 3/2
AU - Ralf Stephan
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19007
UR - https://arxiv.org/abs/2609.19007
AB - Let x_0 be a positive integer, let x_n=⌈ 3x_n−1/2⌉, and let w_n=2x_n+1−3x_n∈{0,1} be the associated word studied by Dubickas; for x_0=1 the orbit is A061419 and w is the minimal word g_3/2 of the rational base number system of Akiyama, Frougny and Sakarovitch. The orbit encodes a real constant K=lim_nx_n(2/3)^n, equal for x_0=1 to ω_3/2=K(3)=1.6222705028…, whose irrationality has been open since 1977. We prove that if w is automatic then K is transcendental; equivalently, an algebraic K forces w to be non-automatic. Further we prove that either the complexity of w exceeds every linear bound or K is irrational. All results are formally verified in the Lean~4 proof assistant, on three cited axioms.
ER -