Geometric Progressions meet Zeckendorf Representations
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911332812455936 |
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| author | Marques, Diego Trojovsky, Pavel |
| author_facet | Marques, Diego Trojovsky, Pavel |
| contents | Motivated by Erdős' 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19586 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric Progressions meet Zeckendorf Representations Marques, Diego Trojovsky, Pavel Number Theory Motivated by Erdős' 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. |
| title | Geometric Progressions meet Zeckendorf Representations |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.19586 |