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.