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arXiv · 2512.19586

Geometric Progressions meet Zeckendorf Representations

Diego Marques·Pavel Trojovsky
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Abstract

Motivated by Erd\H{o}s' ternary conjecture and by recent work of Cui--Ma--Jiang [``Geometric progressions meet Cantor sets'', \textit{Chaos Solitons Fractals} \textbf{163} (2022), 112567.] on intersections between geometric progressions and Cantor-like sets in standard bases, we study the corresponding problem in the Zeckendorf numeration system. We prove that, for any fixed finite set of forbidden binary patterns, any integers $u\ge 1$, $q\ge 2$, and any window size $M$, the set of exponents $n$ for which the Zeckendorf expansion of $u q^n$ avoids the forbidden patterns within its $M$ least significant digits is either finite or ultimately periodic.

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Diego Marques, Pavel Trojovsky. 2025-12-22. Geometric Progressions meet Zeckendorf Representations. https://arxiv.org/abs/2512.19586

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