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.
Highlights
Sequences of the form (ë jaû) jÎN for a >1, known as Beatty sequences, were first studied in the context of the famous problem of covering the set of positive integers by disjoint sequences [1]
Their study is rather technical; they have used for their proof the Zeckendorf representation of integers as a sum of distinct Fibonacci numbers
We show that Beatty sequences (ë jaû) jÎN
Summary
Sequences of the form (ë jaû) jÎN for a >1, known as Beatty sequences, were first studied in the context of the famous problem of covering the set of positive integers by disjoint sequences [1]. In [12] the authors study the self-matching properties of the Beatty sequence (ë jtû) jÎN for t. We show that Beatty sequences (ë jaû) jÎN for quadratic Pisot units a have a similar self-matching property, and for our proof we use a simpler method, based on the cut-and-project scheme. It is interesting to note that Beatty sequences, Fibonacci numbers and the cut-and-project scheme have attracted the attention of physicists in recent years because of their applications for mathematical description of non-crystallographic solids with long-range order, so-called quasicrystals, discovered in 1982 [13]. This necessitates, for an algebraic description of the mathematical model of such a structure, the use of the quadratic field Q(t) Such a model is self-similar with the scaling factor t-1. M, and n are quadratic Pisot units, i.e. they belong to the class of numbers for which the result of Bunder and Tognetti is generalized here
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