Journal of Integer Sequences, Vol. 27 (2024), Article 24.3.2

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

Yohan Hosten
Laboratoire Amiénois de Mathématique
Fondamentale et Appliquée Université de Picardie Jules Verne
33, rue Saint Leu
80000 Amiens


We study probability measures defined by the variation of the sum of digits in the Zeckendorf representation. For r ≥ 0 and dZ, we consider μ(r)(d), the density of integers nN 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(F) = μ(1) where F 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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