A dynamical system approach to the Kakutani-Fibonacci sequence

TL;DR

This paper introduces a dynamical system approach to the Kakutani-Fibonacci sequence, showing that the orbit of the origin under a specific ergodic transformation produces a low discrepancy sequence, linking partition sequences and point distributions.

Contribution

It establishes a novel connection between Kakutani-Fibonacci partitions and ergodic interval exchange transformations, providing a dynamical systems perspective on low discrepancy sequences.

Findings

The orbit of the origin under the Kakutani-Fibonacci transformation forms a low discrepancy sequence.

The Kakutani-Fibonacci sequence of points is generated via an ergodic interval exchange.

The approach links partition refinement sequences to dynamical systems and uniform distribution.

Abstract

In this paper we consider the sequence of Kakutani's $\alpha$-refinements corresponding to the inverse of golden ratio (which we call Kakutani-Fibonacci sequence of partitions) and associate to it an ergodic interval exchange (which we call Kakutani-Fibonacci transformation) using the "cutting-stacking" technique. We prove that the orbit of the origin under this map coincides with a low discrepancy sequence (which we call Kakutani-Fibonacci sequence of points), which has been also considered by other authors.