{"id":"hexagon/2611.00108","pdf":"https://hexagonmath.org/pdf/2611.00108v1","url":"https://hexagonmath.org/abs/2611.00108v1","tags":["msc:11J72","msc:11A55","2-adic integers","3x+1 problem","Conjugacy maps","Diophantine approximation","Factor complexity","Number Theory","Sturmian words","combinatorics on words"],"score":0,"title":"TRANSCENDENCE OF THE 3x+ 1 CONJUGACY MAP ON STURMIAN WORDS","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Elias De Jesús"],"reviews":0,"subject":"math.NT","summary":"Let T be the 3x + 1 map on the 2-adic integers and let Φ : ℤ₂ → ℤ₂ be the Bernstein–Lagarias conjugacy map, which sends a parity vector to the unique 2-adic integer realising it. We prove that Φ(cγ), Φ(1cγ), Φ(0cγ) and Φ of every shift of these words are transcendental for every irrational slope γ < γ* = (3 + √5)/(4 log₂ 3) = 0.8258977…, with no hypothesis whatever on the partial quotients of γ; in particular at the resonance slope γ = log₃ 2. The mechanism is a bridge: by the Bernstein–Lagarias isometry, the 2-adic approximation depth of a periodic shadow equals a longest common prefix, so…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-11-05","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}