Fibonacci Fronts in Iterated Pairwise-Distance Sets

Abstract

For a finite set of integers A, let D(A)=\{|x-y|:x,y\in A\} be its set of distinct pairwise distances, with distance zero included, and iterate A_{k+1}=D(A_k). We determine the complete orbit of two natural three-point families. If n\ge5 is odd and A_0=\{0,2,n\}, then before a collision the orbit is the union of an even prefix and an odd suffix whose lengths are consecutive Fibonacci numbers. The collision produces every odd integer in [1,n], and one further iteration gives the full interval \{0,1,\ldots,n\}. Consequently the exact closure time is the least r\ge3 such that F_r\ge(n-1)/2. An analogous interval-front description is obtained for \{0,1,n\}.