Recurrence with prescribed number of residues

Abstract

In this paper we show that for every positive integer m there exist positive integers x1,x2,M such that the sequence (xn)n=1∞ defined by the Fibonacci recurrence xn+2=xn+1+xn, n=1,2,3,…, has exactly m distinct residues modulo M. As an application we show that for each integer m⩾2 there exists ξ∈R such that the sequence of fractional parts {ξφn}n=1∞, where φ=(1+5)/2, has exactly m limit points. Furthermore, we prove that for no real ξ≠0 it has exactly one limit point.