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

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