On Problem 5 of Vitorino and Vukusic: $\sup_{k\ge 0} B(2^k\alpha)\ge 12$
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…
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