Fibonacci Expansions and "f-adic" Integers

A Fibonacci expansion of a nonnegative integer n is an expression of n as a sum of Fibonacci numbers Fk with k > 2. It may be thought of as a partition of n into Fibonacci parts. The most commonly studied such expansion is the unique one in which the parts are all distinct and no two consecutive Fibonacci numbers appear. C. G. Lekkerkerker first showed this expansion was unique in 1952 [5]. There is also a unique dual form of this expansion in which no two consecutive Fibonacci numbers not exceeding n do not occur in the expansion [2]. Lekkerkerker's expansion is the only one I refer to in the remainder of this paper; from now on, I will call it the Fibonacci expansion of n, or fib(w) (I will give a precise definition in Part II). The Fibonacci expansion of nonnegative integers is similar in many ways to a fixed-base expansion (in fact, in some sense, it may be thought of as a base-r expansion, where r = y(l + V5)« 1.61803 is the golden mean). First, in each case there are both "top-down" and "bottom-up" algorithms for obtaining the expansion of a nonnegative integer (see [3], pages 281-282). Second, there are mechanical rules for adding the expansions of two or more nonnegative integers [1]. Third, each case may be generalized by defining infinite expansions (p-adic or "F-adic" integers), both of which have interesting algebraic properties. One should be warned, however, that this analogy has its limitations. For instance, the /?-adic integers form a ring, but the F-adic integers do not. My main result in this paper is that there is a 1-1 correspondence between the F-adic integers and the points on a circle, and that both of these sets share some important geometric properties.