LAUSR

Abstract In this paper we show that for every positive integer m there exist positive integers x 1 , x 2 , M such that the sequence ( x n ) n = 1 ∞ defined by the Fibonacci recurrence x n + 2 = x n + 1 + x n , 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 { ξ φ n } 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.