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On the Variation of the Sum of Digits in the Zeckendorf Representation: An Algorithm to Compute the Distribution and Mixing Properties
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Yohan Hosten

Laboratoire Amiénois de Mathématique

Fondamentale et Appliquée Université de Picardie Jules Verne

33, rue Saint Leu

80000 Amiens

France

**Abstract:**

We study probability measures defined by the variation of the sum of
digits in the Zeckendorf representation. For *r* ≥ 0 and *d* ∈ **Z**, we
consider μ^{(r)}(*d*), the density of integers *n* ∈ **N** for which the sum
of digits increases by *d* when *r* is added to *n*. We give a probabilistic
interpretation of μ^{(r)} via the dynamical system provided by the odometer
of Zeckendorf-adic integers and its unique invariant measure. We give
an algorithm for computing μ^{(r)} and we prove the exponential decay of
μ^{(r)}(*d*) as *d* → -∞, as well as the formula
μ^{(Fℓ)} = μ^{(1)}
where *F*_{ℓ}
is a term of 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.

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Received October 13 2023; revised versions received October 18 2023; January 19 2024; February 23 2024.
Published in *Journal of Integer Sequences*,
February 24 2024.

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