The converse of Zeckendorf theorems for real numbers

Abstract

Every positive integer can be written as a sum of distinct nonadjacent terms of the Fibonacci sequence; it is known as Zeckendorf's Theorem.  It is less well known that there exists a similar theorem for the real numbers in the open interval $(0,1)$.  The converse of the theorem for generalized expansions, called periodic Zeckendorf expansions for the real numbers, is proved in the author's earlier paper. In this paper, we introduce a new approach to this converse problem.