The weak converse of Zeckendorf's Theorem

TL;DR

This paper explores the weak converse of Zeckendorf's Theorem, investigating monotone sequences with unique Zeckendorf-like decompositions, generalizing conditions, and extending results to real numbers and p-adic integers.

Contribution

It introduces a generalization of Zeckendorf conditions, proves theorems for these conditions, and extends the framework to real numbers and p-adic integers.

Findings

Established the weak converse for generalized Zeckendorf conditions

Extended results to real numbers in (0,1)

Extended results to p-adic integers

Abstract

By Zeckendorf's Theorem, every positive integer is uniquely written as a sum of non-adjacent terms of the Fibonacci sequence, and its converse states that if a sequence in the positive integers has this property, it must be the Fibonacci sequence. If we instead consider the problem of finding a monotone sequence with such a property, we call it the weak converse of Zeckendorf's theorem. In this paper, we first introduce a generalization of Zeckendorf conditions, and subsequently, Zeckendorf's theorems and their weak converses for the general Zeckendorf conditions. We also extend the generalization and results to the real numbers in the interval $(0,1)$, and to $p$-adic integers.