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

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.