Self-Matching Properties of Beatty Sequences

Z. Masáková, E. Pelantová

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 Gi of a generalized Fibonacci sequence almost always causes the translation of the value of the function by Gi=1. More precisely, for fixed i ϵ ℕ, we have ⌊β(j+Gi)⌋ = ⌊βj⌋ + Gi=1, where j ϵ Ui. We determine the set Ui of mismatches and show that it has a low frequency, namely βi.


Keywords


Beatty sequences; Fibonacci numbers; cut-and-project scheme

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ISSN 1210-2709 (Print)
ISSN 1805-2363 (Online)
Published by the Czech Technical University in Prague