{"id":"hexagon/2611.00009","pdf":"https://hexagonmath.org/pdf/2611.00009v1","url":"https://hexagonmath.org/abs/2611.00009v1","tags":["math.DS","msc:11J61","msc:11J13","msc:11A55","msc:37B10","Raney transducers","computer-assisted proof","continued fractions","ideal classes","p-adic Littlewood conjecture"],"score":0,"title":"On Problem 5 of Vitorino and Vukusic: $\\sup_{k\\ge 0} B(2^k\\alpha)\\ge 12$","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["David Niedbala Giraudin"],"reviews":0,"subject":"math.NT","summary":"For an irrational $\\alpha$ let $B(\\alpha)=\\limsup_n a_n(\\alpha)$ be the upper limit of its partial quotients. Vitorino and Vukusic proved $\\sup_{k\\ge 0}B(2^k\\alpha)\\ge 5$ for every irrational $\\alpha$ and asked, as their Problem 5, for an improvement. We prove $\\sup_{k\\ge 0}B(2^k\\alpha)\\ge 12$. The proof is a finite computation organised around an invariant $N(C)$: the least number of doubling levels at which the recurrent kernel of a synchronous product of Raney transducers becomes a finite union of pure cycles. We show that $N(C)$ is intrinsic, that $N(C)<\\infty$ for every $C$ would imply…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}