bibkey: carbone2012dynamical authors: “Ingrid Carbone; Maria Rita Iacò; Aljoša Volčič” year: 2012 title: “A dynamical system approach to the Kakutani-Fibonacci sequence” doi: null url: “https://arxiv.org/abs/1211.0708v1” claim: “Definition 1 refines every longest interval at a fixed ratio; for the inverse golden ratio the partition lengths are consecutive powers and their counts satisfy the Fibonacci recurrence.” strata_touched: [] license: “citation-only” triage: “anchor”

Kakutani–Fibonacci partition supplier

The primary version is arXiv:1211.0708v1. Definition 1 in §1, PDF p.2, defines Kakutani’s $\alpha$-refinement of a partition of $[0,1)$ by splitting all intervals of maximal length into parts proportional to $\alpha$ and $1-\alpha$. Shorter intervals are retained.

In §1, PDF p.4, the choice $\alpha =(\sqrt{5}-1)/2$ satisfies ${\alpha }^2+\alpha =1$. The successive partitions have long intervals of length ${\alpha }^n$ and short intervals of length ${\alpha }^{n+1}$; their long, short and total counts satisfy the Fibonacci recurrence with the initial values stated there. These facts are literature-attested ingredients.

The single-lineage partition volume consumes Definition 1 and the two-length relation in §§5.2–5.3. Its identification with the labelled FIB process requires a single initial $\beta$, the golden ratio, and the proof that precisely the $\beta$ intervals are longest. An initial $\alpha$ adds a relabelling step before any split.

The paper also constructs an interval exchange and proves ergodic and discrepancy results. Those results are not suppliers for the volume’s fixed read-point model: equality of partition lengths alone does not supply the paper’s point ordering or interval exchange. No such transfer is claimed.