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Geometric Progressions meet Zeckendorf Representations

Diego Marques, Pavel Trojovsky

TL;DR

The paper studies how geometric progressions interact with Zeckendorf representations under local digit constraints, mirroring prior work in standard bases. It encodes carry propagation of multiplication by $q$ as a finite-state transducer in the Fibonacci (Zeckendorf) system and analyzes the evolution of the least-significant-digit window. The main result shows that for any fixed window length $M$ and forbidden-pattern family $$, the set of exponents $n$ for which $uq^n$ avoids the patterns in the first $M$ digits is finite or ultimately periodic, with an effective procedure to compute the period. The work demonstrates a fully finite-state approach to local digit constraints in Zeckendorf numeration, clarifying the local dynamics of Fibonacci-based arithmetic and highlighting the distinction between local (window) and global (full expansion) avoidance problems. It extends the framework of geometric-progressions-with-digit-restrictions to the Zeckendorf setting, connecting automata theory with number-theoretic digit constraints.

Abstract

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.

Geometric Progressions meet Zeckendorf Representations

TL;DR

The paper studies how geometric progressions interact with Zeckendorf representations under local digit constraints, mirroring prior work in standard bases. It encodes carry propagation of multiplication by as a finite-state transducer in the Fibonacci (Zeckendorf) system and analyzes the evolution of the least-significant-digit window. The main result shows that for any fixed window length and forbidden-pattern family , the set of exponents for which avoids the patterns in the first digits is finite or ultimately periodic, with an effective procedure to compute the period. The work demonstrates a fully finite-state approach to local digit constraints in Zeckendorf numeration, clarifying the local dynamics of Fibonacci-based arithmetic and highlighting the distinction between local (window) and global (full expansion) avoidance problems. It extends the framework of geometric-progressions-with-digit-restrictions to the Zeckendorf setting, connecting automata theory with number-theoretic digit constraints.

Abstract

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 , , and any window size , the set of exponents for which the Zeckendorf expansion of avoids the forbidden patterns within its least significant digits is either finite or ultimately periodic.
Paper Structure (7 sections, 3 theorems, 37 equations, 3 figures, 1 table)

This paper contains 7 sections, 3 theorems, 37 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['thm:main_window']}
  3. Automata-theoretic dictionary
  4. Auxiliary results
  5. The proof
  6. An Illustrative Example
  7. Concluding remarks

Key Result

Theorem 1.1

Let $u\ge 1$, $q\ge 2$ be integers, and let $\mathcal{F}$ be a finite family of non-empty binary words. Let $L=\max\{|f|:f\in\mathcal{F}\}$ and fix $M\ge L$. Then the set is either finite or ultimately periodic. That is, there exist integers $n_0\ge 0$ and $p\ge 1$ such that Moreover, one can effectively compute such a pair $(n_0,p)$ and decide whether $S_u^{(M)}$ is finite or infinite. In the f

Figures (3)

  • Figure 1: Schematic excerpt of the $q=2$ multiplication transducer (not all states/transitions shown). It reads $\widetilde{Z}(N)=\mathrm{rev}(Z(N))\#^\omega$ LSD-first and outputs $\widetilde{Z}(2N)$; edges are labeled by input/output pairs. The dashed arrow indicates omitted states.
  • Figure 2: Illustrative excerpt of the window dynamics for $q=2$ and $M=3$. Vertices are windows $w\in\{0,1,\#\}^3$, coloured green if $w$ avoids $101$ and red otherwise ($\mathcal{F}=\{101\}$). Thick arrows indicate a sample forward trajectory from $w_0=\mathrm{pref}_3(\widetilde{Z}(1))=\texttt{1\#\#}$.
  • Figure 3: Orbit structure for $(w_n)$ with $u=1$, $q=2$, $\mathcal{F}=\{101\}$ and window $M=5$. The window orbit has a preperiod of length $29$ followed by a cycle of length $4$. In this instance the cycle consists entirely of rejecting states, hence $S_1^{(5)}$ is finite.

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Remark 3.1