Self-Matching Properties of Beatty Sequences

We study the selfmatching properties of Beatty sequences, in particular of the graph of the function $\lfloor j\beta\rfloor $ against $j$ for every quadratic unit $\beta\in(0,1)$. We show that translation in the argument by an element $G_i$ of generalized Fibonacci sequence causes almost always the translation of the value of function by $G_{i-1}$. More precisely, for fixed $i\in\N$, we have $\bigl\lfloor \beta(j+G_i)\bigr\rfloor = \lfloor \beta j\rfloor +G_{i-1}$, where $j\notin U_i$. We determine the set $U_i$ of mismatches and show that it has a low frequency, namely $\beta^i$.


Introduction
Sequences of the form ⌊jα⌋ j∈N for α > 1, now known as Beatty sequences, have been first studied in the context of the famous problem of covering the set of positive integers by disjoint sequences [1]. Further results in the direction of the so-called disjoint covering systems are due to [5,7,14] and others. Other aspects of Beatty sequences were then studied, such as their generation using graphs [4], their relation to generating functions [9,10], their substitution invariance [8,11], etc. A good source of references on Beatty sequences and other related problems can be found in [2,13].
In [3] the authors study the self-matching properties of the Beatty sequence ⌊jτ ⌋ j∈N for τ = 1 2 ( √ 5 − 1), the golden ratio. Their study is rather technical; they have used for their proof the Zeckendorf representation of integers as a sum of distinct Fibonacci numbers. The authors also state an open question whether the results obtained can be generalized to other irrationals than τ . In our paper we answer this question in the affirmative. We show that Beatty sequences ⌊jα⌋ j∈N for quadratic Pisot units α have analogical self-matching property, and for our proof we use a simpler method, based on the cut-and-project scheme.
It is interesting to mention that Beatty sequences, Fibonacci numbers and cutand-project scheme attracted the attention of physicists in recent years because of their applications for mathematical description of non-crystallographic solids with long-range order, the so-called quasicrystals, discovered in 1982 [12]. The first observed quasicrystals revealed crystallographically forbidden rotational symmetry of order 5. This necessitates, for the algebraic description of the mathematical model of such a structure, the use of the quadratic field Q(τ ). Such a model is self-similar with the scaling factor τ −1 . Later, one observed existence of quasicrystals with 8 and 12-fold rotational symmetries, corresponding to mathematical models with selfsimilar factors µ −1 = 1 + √ 2 and ν −1 = 2 + √ 3. Note that all τ , µ, and ν are quadratic Pisot units, i.e. belong to the class of numbers for which the result of Bunder and Tognetti is generalized here.

Quadratic Pisot units and cut-and-project scheme
The self-matching properties of the Beatty sequence ⌊jτ ⌋ j∈N are best displayed on the graph of ⌊jτ ⌋ against j ∈ N. Important role is played by the Fibonacci numbers, The result of [3] states that except isolated mismatches of frequency τ i , namely at points j = kF i+1 + ⌊kτ ⌋F i . Our aim is to show a very simple proof of the mentioned results that is valid for all quadratic units β ∈ (0, 1). Every such unit is a solution of the quadratic equation The considerations will slightly differ in the two cases.
(a) Let β ∈ (0, 1) satisfy β 2 + mβ = 1 for m ∈ N. The algebraic conjugate of β, i.e. the other root of the equation, satisfies β ′ < −1. We define the generalized Fibonacci sequence It is easy to show by induction that for i ∈ N, we have In this case, we have for i ∈ N The proof we give here is based on the algebraic expression for one-dimensional cut-and-project sets [6]. Let V 1 , V 2 be straight lines in R 2 determined by vectors (β, −1) and (β ′ , −1), respectively. The projection of the square lattice Z 2 on the line V 1 along the direction of V 2 is given by . For the description of the projection of Z 2 on V 1 it suffices to consider the set The integral basis of this free abelian group is (1, β ′ ), and thus every element x of Z[β ′ ] has a unique expression in this base. We will say that a is the rational part of x = a + bβ ′ and b is its irrational part. Since β ′ is a quadratic unit, Z[β ′ ] is a ring and, moreover, it satisfies A cut-and-project set is the set of projections of points of Z 2 to V 1 , that are found in a strip of bounded width, parallel to the straight line V 1 . Formally, for a bounded interval Ω we define Note that a + bβ ′ corresponds to the projection of the point (a, b) to the straight line V 1 along V 2 , whereas a + bβ corresponds to the projection of the same lattice point to V 2 along V 1 .
Among the simple properties of cut-and-project sets that we use here are where the latter is a consequence of (6). If the interval Ω is of unit length, one can derive directly from the definition a simpler expression for Σ(Ω). In particular, we have where we use that the condition 0 ≤ a+bβ < 1 is satisfied if and only if a = ⌈−bβ⌉ = −⌊bβ⌋. Let us mention that the above properties of one-dimensional cut-and-project sets, and many others, are explained in the review article [6].
3 Self-matching property of the graph ⌊jβ⌋ against j Important role in the study of self-matching properties of the graph ⌊jβ⌋ against j is played by the generalized Fibonacci sequence (G i ) i∈N , defined by (2) and (4), respectively. It turns out that shifting the argument j of the function ⌊jβ⌋ by the integer G i results in shifting the value by G i−1 , except of isolated mismatches with low frequency. The first proposition is an easy consequence of the expressions of β i as an element of the ring Z[β] in the integral basis 1, β, given by (3) and (5). Theorem 1. Let β ∈ (0, 1) satisfy β 2 + mβ = 1 and let (G i ) ∞ i=0 be defined by (2). Let i ∈ N. Then for j ∈ Z we have The frequency of integers j, for which the value ε i (j) is non-zero, is equal to Proof. The first statement is trivial. For, we have The frequency ̺ i is easily determined in the proof of Theorem 2.
In the following theorem we determine the integers j, for which ε i (j) is non-zero. From this, we easily derive the frequency of such mismatches.
Theorem 2. With the notation of Theorem 1, we have Before starting the proof, let us mention that for i even, the set U i can be written simply as U i = kG i+1 + ⌊kβ⌋G i k ∈ Z . For i odd, the element corresponding to k = 0 is equal to −G i instead of 0. The distinction according to parity of i is necessary here, since unlike the paper [3], we determine the values of ε i (j) for j ∈ Z, not only j ≥ 1.
Proof. It is convenient to distinguish two cases according to the parity of i.
• First let i be even. It is obvious from (7), that ε i (j) ∈ {0, −1} and Let us denote by M the set of all such j, Therefore M is formed by the irrational parts of the elements of the set Separating the irrational part we obtain where we have used the equations β ′ 2 + mβ ′ = 1 and mG i + G i−1 = G i+1 .
• Let now i be odd. Then from (7), ε i (j) ∈ {0, 1} and Let us denote by M the set of all such j, Therefore M is formed by the irrational parts of elements of the set Separating the irrational part we obtain where we have used the equation Let us recall that the Weyl theorem [15] says that numbers of the form αj −⌊αj⌋, j ∈ Z, are uniformly distributed in (0, 1) for every irrational α. Therefore the frequency of those j ∈ Z that satisfy αj − ⌊αj⌋ ∈ I ⊂ (0, 1) is equal to the length of the interval I. Therefore one can derive from (8) and (9) that the frequency of mismatches (non-zero values ε i (j)) is equal to β i , as stated by Theorem 1.
If β ∈ (0, 1) is the quadratic unit satisfying β 2 − mβ = −1, then the considerations are even simpler, because the expression (5) does not depend on the parity of i. We state the result as the following theorem.
Then for j ∈ Z we have The density of the set U i of mismatches is equal to β i .
Proof. The proof follows the same lines as proofs of Theorems 1 and 2.

Conclusions
One-dimensional cut-and-project sets can be constructed from Z 2 for every choice of straight lines V 1 , V 2 , if the latter have irrational slopes. However, in our proof of the self-matching properties of the Beatty sequences we strongly use the algebraic ring structure of the set Z[β ′ ], and its scaling invariance with the factor β ′ , namely β ′ Z[β] = Z[β ′ ]. For that, β ′ being quadratic unit is necessary. However, it is plausible, that even for other irrationals α, some self-matching property is displayed by the graph ⌊jα⌋ against j. For showing that, other methods would be necessary.