On the variation of the sum of digits in the Zeckendorf representation: an algorithm to compute the distribution and mixing properties

TL;DR

This paper investigates the distribution of digit sum variations in Zeckendorf representations, providing algorithms and probabilistic interpretations, and demonstrating mixing properties of the associated dynamical system.

Contribution

It introduces a novel algorithm to compute the distribution of digit sum changes and analyzes the mixing properties of Zeckendorf-adic integers.

Findings

Derived an explicit formula for the distribution (r)

Established the invariance (F_{\u03bb}) = (1) for Fibonacci indices

Showed the sequence of block actions exhibits mixing behavior

Abstract

We study probability measures defined by the variation of the sum of digits in the Zeckendorf representation. For $r\ge 0$ and $d\in\mathbb{Z}$, we consider $\mu^{(r)}(d)$ the density of integers $n\in\mathbb{N}$ for which the sum of digits increases by $d$ when $r$ is added to $n$. We give a probabilistic interpretation of $\mu^{(r)}$ via the dynamical system provided by the odometer of Zeckendorf-adic integers and its unique invariant measure. We give an algorithm for computing $\mu^{(r)}$ and we deduce a control on the tail of the negative distribution of $\mu^{(r)}$, as well as the formula $\mu^{(F_{\ell})} = \mu^{(1)}$ where $F_{\ell}$ is a term in the Fibonacci sequence. Finally, we decompose the Zeckendorf representation of an integer $r$ into so-called "blocks" and show that when added to an adic Zeckendorf integer, the successive actions of these blocks can be seen as a sequence of mixing random variables.