The weak converse of Zeckendorf's Theorem

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.