Self-Matching Properties of Beatty Sequences

Abstract

We study the selfmatching properties of Beatty sequences, in particular of the graph of the function  ⌊ jβ ⌋ against j for every quadratic unit βϵ (0,1). We show that translation in the argument by an element G_i of a generalized Fibonacci sequence almost always causes the translation of the value of the function by G_i=1. More precisely, for fixed i ϵ ℕ, we have ⌊β(j+G_i)⌋ = ⌊βj⌋ + G_i=1, where j ϵ Ui. We determine the set U_i of mismatches and show that it has a low frequency, namely βⁱ.