Geometric Progressions meet Zeckendorf Representations

TL;DR

This paper explores the behavior of Zeckendorf representations of scaled integers, proving that the set of exponents avoiding certain digit patterns is either finite or eventually repeats, linking geometric progressions with number representations.

Contribution

It establishes a dichotomy for pattern-avoiding Zeckendorf expansions of scaled integers, extending previous work on geometric progressions and Cantor sets.

Findings

The set of exponents avoiding forbidden patterns is finite or ultimately periodic.

The result applies to any fixed finite set of forbidden binary patterns.

The proof connects Zeckendorf representations with properties of geometric progressions.

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.