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

David Niedbala Giraudin

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

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory math.DS — Dynamical Systems

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

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 the

2

-adic Littlewood conjecture, and that

N(C)=\infty

is equivalent to a rigidity failure; we compute

N(C)=1,1,3,5,5,8,15,21,29,30,31

for

C=1,\dots,11

. In the language of Badziahin's set LMad this settles a level-bounded form of his Problem B for

C\le 11

. The cycles which survive are ideal classes of discriminant

D\equiv 1 \pmod 8

whose whole

\mathfrak{p}_2

-orbit has partial quotients at most

C

; we list them, and check the lists by an independent arithmetic route, in both directions for

D\le 10^5

. The result concerns the upper limit

B

only. On the side of the supremum

M

it gives

\sup_k M(2^k\alpha)\ge 12

, which is weaker than what is known: Vitorino and Vukusic prove

15

, and Badziahin's bound

1/25

yields

24

. Two self-contained Python files, without external dependency, reproduce the table and the lists of discriminants; the table up to

C=8

takes about fourteen seconds.

Provenance statement

The research, the computations and the writing were carried out by the author with Claude (Anthropic). All computational claims are reproduced by the two ancillary Python files in anc/. The bibliography was checked against the original sources. The note has not yet been reviewed by professionals.

Tools used

Anthropic: Claude Opus

References

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Version history

v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 01, 2026