SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS
December 2, 2013
arXiv:1312.0653v1 [math.MG] 2 Dec 2013
SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS TOMÁŠ HEJDA AND EDITA PELANTOVÁ Abstract. For q ∈ R, q > 1, Erdős, Joó and Komornik study distances of the consecutive points in the set n nX o X m (q) = aj q j : n ∈ N, ak ∈ {0, 1, . . . , m} . j=0
The Pisot numbers play a crucial role for properties of X m (q). We follow work of Zaïmi who consideres X m (γ) with γ ∈ C \ R and γγ > 1. We show that for any non-real γ and m < γγ − 1, the set X m (γ) is not relatively dense in the complex plane. For a class of cubic complex Pisot units γ and m > γγ − 1 we deduce that X m (γ) is uniformly discrete and relatively dense, i.e., X m (γ) is a Delone set. For γ the complex root of Y 3 + Y 2 + Y − 1 we determine two parameters of the Delone set X m (γ) which are analogous to minimal and maximal distances for the real case X m (q).
1. Introduction In [EJK90, EJK98], Erdős, Joó and Komornik studied the set n nX o X m (β) := aj β j : n ∈ N, ak ∈ {0, 1, . . . , m} , j=0
where β > 1. Since this set has no accumulation points, we can find an increasing sequence 0 = x0 < x1 < x2 < · · · < xk < · · · such that X m (β) = {xk : k ∈ N}. The research of Erdős et al. aims to describe distances between consecutive points of X m (β), i.e., the sequence (xk+1 − xk )k∈N . The properties of this sequence depend on the value m ∈ N. It is easy to show that when m ≥ β − 1, we have xk+1 − xk ≤ 1 for all k ≥ 0; and when m < β − 1, the distances xk+1 − xk can be arbitrarily large. The properties of X m (β) depend on β being a Pisot number (i.e., an algebraic integer > 1 such that all its Galois conjugates are in modulus < 1). Bugeaud [Bug96] showed that `m (β) := lim inf (xk+1 − xk ) > 0 for all m ∈ N k→∞
if and only if the base β is a Pisot number. Recently, Feng [Fen13] proved a stronger result that the bound β − 1 for the alphabet size is crucial. In particular, `m (β) = 0 if and only if m > β − 1 and β is not a Pisot number. Therefore, the case β Pisot and m > β − 1 has been further studied. From the approximation property of Pisot numbers we know that for a fixed β and m > β − 1 2010 Mathematics Subject Classification. Primary 11A63, 52C23, 52C10; Secondary 11H99, 11-04.
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T. HEJDA AND E. PELANTOVÁ
the sequence (xk+1 − xk ) takes only finitely many values. Feng and Wen [FW02] used this fact to show that the sequence of distances (xk+1 − xk ) is substitutive, roughly speaking, can be generated by a system of rewriting rules over a finite alphabet. This allows, for a fixed β and m, to determine values of all distances (xk+1 − xk ) and subsequently the value of `m (β). An algorithm for obtaining the minimal distance `m (β) for certain β was as well proposed by Borwein and Hare [BH02]. The first formula which determines the value of `m (β) for all m at once appeared in 2000: Komornik, Loreti and Pedicini [KLP00] studied the base golden mean. The generalization of this result to all quadratic Pisot units was provided by Takao Komatsu [Kom02] in 2002. To the best of our knowledge, the value of (1.1)
Lm (β) := lim sup(xk+1 − xk ) k→∞
for all m is only known for the base Golden mean, due to Borwein and Hare [BH03]. Of course, for a given m, the value of Lm (β) can be computed using [FW02]. Zaïmi [Zaï04] was interested in a complementary question: Fix the alphabet size, i.e., the maximal digit m, and look for the extremal values of `m (β) where β runs through the Pisot numbers in (m, m + 1). Zaïmi showed that `m (β) is maximized for certain quadratic Pisot numbers. Besides that, Zaïmi started to study the set X m (γ) where he considered γ a complex number of modulus > 1, and he put (1.2) `m (γ) := inf |x − y| : x, y ∈ X m (γ), x 6= y . He proved an analogous result to the one for real bases by Bugeaud, namely that `m (γ) > 0 for all m if and only if γ is a complex Pisot number, which is defined as a non-real algebraic integer of modulus > 1 whose Galois conjugates except its complex conjugate are in modulus < 1. In the complex plane, `m (γ) and Lm (γ) cannot be defined as simply as in the real case since we have no natural ordering of the set X m (γ) in C. To overcome this, we will inspire by notions used in the definition of Delone sets. We say that a set Σ is: • uniformly discrete if there exists d > 0 such that |x − y| ≥ d for all distinct x, y ∈ Σ; • relatively dense if there exists D > 0 such that every ball B(x, D/2) of radius D/2 contains a point from Σ. A set that is both uniformly discrete and relatively dense is called Delone set. Clearly, if `m (γ) as given by (1.2) is positive, then X m (γ) is uniformly discrete and `m (γ) is the maximal d in the definition of uniform discreteness. Let us define Lm (γ) := inf D > 0 : B(x, D/2) ∩ X m (γ) 6= ∅ for all x ∈ C . In particular, Lm (γ) = +∞ if and only if X m (γ) is not relatively dense. The question for which pairs (γ, m) the set X m (γ) is uniformly discrete, and for which (γ, m) it is relatively dense is far from being solved. We provide a necessary condition for relative denseness and we show that in certain cases, it is sufficient as well: Theorem 1.1. Let γ ∈ C be a non-real number in modulus > 1.
SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS
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(1) If m ≤ γγ − 1, then X m (γ) is not relatively dense. (2) [Zaï04] If m > γγ − 1 and γ is not an algebraic number, then X m (γ) is not uniformly discrete. The aim of this article is to study the sets X m (γ) simultaneously for all m ∈ N, for a certain class of cubic complex Pisot numbers with a positive conjugate γ 0 . For such γ the Rényi expansions in the base β := 1/γ 0 have nice properties which will be crucial in the proofs. When the base β = 1/γ 0 satisfies so-called Property (F), we show that for all sufficiently large m the set X m (γ) ⊆ C is a cut-and-project set; roughly speaking, X m (γ) is formed by projections of points from the lattice Z3 which lie in a sector bounded by two parallel planes in R3 , see Theorem 4.1. From that easily follows the asymptotic behaviour of `m (γ) and Lm (γ), namely: √ √ (1.3) `m (γ) = Θ(1/ m) and Lm (γ) = Θ(1/ m), √ √ √ where f (m) = Θ(1/ m) means that K1 / m ≤ f (m) ≤ K2 / m for some positive constants K1,2 . Any cut-and-project set Σ has finite local complexity, which means that there are only finitely many types of arrangements of close neighborhoods of points of Σ. More formally, for any % > 0 the set of patches (Σ − x) ∩ B(0, %) : x ∈ Σ is finite. In particular, it means that there are only finitely many Voronoi cells determined by the set Σ. The method of inspection of Voronoi cells for a specific cut-and-project set, as established by Masáková, Patera and Zich [MPZ03a, MPZ03b, MPZ05], enables us to give a general formula for both `m (γ) and Lm (γ). In the case that γ is the complex Tribonacci constant, i.e., the complex root of Y 3 + Y 2 + Y − 1, we get: Theorem 1.2. Let γ = γT ≈ −0.771 + 1.115i be the complex root of the polynomial Y 3 + Y 2 + Y − 1 and m ∈ N. Let k ∈ Z be the maximal integer such that m ≥ k (1 − γ 0 ) γ10 , where γ 0 is the real Galois conjugate of γ. Then we have r 1 − (γ 0 )2 3−k −k (1.4) `m (γ) = |γ| and Lm (γ) = 2 |γ| . 3 − (γ 0 )2 The article is organized as follows. In Section 2, we recall certain notions from the theory of β-expansions. Section 3 provides the proof of the 1st part of Theorem 1.1. In Section 4 we prove that X m (γ) is a cut-and-project set in certain cases. Section 5 describes the algorithms for computing `m (γ) and Lm (γ). These algorithms are applied to the complex Tribonacci number in Section 6, providing the proof of Theorem 1.2. In Section 7 we compute another characteristic of X m (γ) that is based on Delone tilings. Comments and open problems are in Section 8. All computations were carried out in C++ and in Sage [Sage]. The pictures were drawn using Tik Z [Tik Z]. 2. Preliminaries Let us recall some facts concerning β-expansions. For a real base β > 1, and for a number x ≥ 0, there exist unique N ∈ Z and unique integer coefficients
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aN , aN −1 , aN −2 , . . . such that aN 6= 0 and 0≤x−
N X
aj β j < β n
for all n ≤ N .
j=n
The string aN aN −1 · · · a1 a0 .a−1 a−2 · · · is called Rényi expansion of x in the base β. We immediatelly see that aj ∈ {0, . . . , bβc}. If only finitely many aj’s are nonzero, we speak about finite Rényi expansion of x. The set of numbers x ∈ R such that |x| has finite Rényi expansion is denoted Fin(β). We say that β > 1 satisfies Property (F) if Fin(β) is an algebraic ring, i.e., Fin(β) = Z[1/β], where Z[y] denotes as usual the integer combinations of powers of y. We will widely use the algebraic properties of cubic complex Pisot numbers γ. Such γ has two Galois conjugates. One of them is the complex conjugate γ. The second one is real and of modulus < 1, we will denote it γ 0 ; we have either −1 < γ 0 < 0 or 0 < γ 0 < 1. In general, for z ∈ Q(γ) we denote by z 0 ∈ Q(γ 0 ) ⊂ R its image under the Galois isomorphism that maps γ 7→ γ 0 . When γ is a unit (i.e., the absolute term of its minimal polynomial is ±1), we know that Z[1/γ] = Z[γ] = γZ[γ]. Akiyama [Aki00] described the real cubic units having Property (F) in terms of the coefficients of the minimal polynomial. Combining his result and Cardano’s formula we get that non-real γ is a cubic complex Pisot unit such that its real conjugate satisfies γ 0 > 0 and β := 1/γ 0 has Property (F) if and only if γ is a root of Y 3 + bY 2 + aY − 1, where a, b ∈ Z satisfy (2.1)
− 1 ≤ b ≤ a + 1,
18ab + 4a3 − a2 b2 − 4b3 + 27 > 0,
(a, b) 6= (1, −1).
In particular, the complex Tribonacci constant γT ≈ −0.771 + 1.115i (the root of Y 3 + Y 2 + Y − 1) falls into this scheme, and more generally, the complex roots of polynomials Y 3 + bY 2 + aY − 1 for b = 0, ±1 and a ≥ 1, with the exception (a, b) = (1, −1). 3. Proof of Theorem 1.1 We prove the first part of Theorem 1.1. We cannot easily follow the lines of the proof of the result for the real case (i.e., that m < β − 1 implies Lm (β) = +∞). In the proof of the theorem, the following ‘folklore’ lemma about the asymptotic density of relatively dense sets will be used: Lemma 3.1. Let Σ ⊂ C be a set without accumulation points. Suppose Σ is relatively dense. Then # Σ ∩ B(0, r) > 0. (3.1) lim inf r→∞ r2 Proof. Since Σ is relatively dense, there exists λ > 0 such that every square in C with side λ contains a point of Σ. Therefore every cell of the lattice λZ[i] = 2 {λa + iλb : a,√ b ∈ Z} contains a point of Σ. Since B(0, r) contains at least n cells, where n = r 2/λ , we get √ 2 # Σ ∩ B(0, r) r 2/λ 2 lim inf ≥ lim inf = 2 > 0. r→∞ r→∞ r2 r2 λ
SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS
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Proof of Theorem 1.1, 1st statement. For simplicity, we denote Σ := X m (γ). First, we will show that for any r ≥ m we have Σ ∩ B 0, |γ|r − m ⊆ γ Σ ∩ B(0, r) + {0, . . . , m} and therefore (3.2)
# Σ ∩ B(0, |γ|r − m) ≤ (m + 1)# Σ ∩ B(0, r) . Pk To prove this, consider x = j=0 aj γ j with aj ∈ {0, . . . , m} and such that |x| ≤ Pk |γ|r − m. Then y := (x − a0 )/γ = j=1 aj γ j−1 ∈ Σ and |y| ≤ (|x| + a0 )/|γ| ≤ (|γ|r − m + m)/|γ| = r. Since x = γy + a0 , the inclusion is valid. Second, let us define recurrently r0 := m and rk+1 := |γ|rk − m. Clearly rk → ∞ whence rk+1 /rk → γ. Put nk := #(Σ ∩ B(0, rk ))/rk2 . Then (3.2) leads to nk+1 (m + 1)rk2 k→∞ m + 1 ≤ −−−−→ 2 < 1. 2 nk rk+1 |γ| This implies limk→∞ nk = 0, and subsequently lim inf r→∞ #(Σ ∩ B(0, r))/r2 = 0. Therefore the set Σ = X m (γ) is not relatively dense by Lemma 3.1. 4. Cut-and-project sets versus X m (γ) A cut-and-project scheme in dimension d + e consists of two linear maps Ψ : Rd+e → Rd and Φ : Rd+e → Re satisfying: (1) Ψ(Rd+e ) = Rd and restriction of Ψ to the lattice Zd+e is injective; (2) the set Φ(Zd+e ) is dense in Re . Let Ω ⊂ Re be a nonempty bounded set such that its closure equals the closure of its interior, i.e., Ω = Ω◦ . Then the set Σ(Ω) := Ψ(v) : v ∈ Zd+e , Φ(v) ∈ Ω ⊆ Rd is called cut-and-project set with acceptance window Ω. Cut-and-project sets can be defined in a slightly more general way, c.f. [Moo97]. It is well known that Σ(Ω) is a Delone set with finite local complexity. Moreover, in case e = 1, the form of acceptance window Ω = [l, r) or Ω = (l, r] guarantees that Σ(Ω) is repetitive, i.e., for every x ∈ Σ(Ω) and % > 0 the patch (Σ(Ω) − x) ∩ B(0, %) occurs infinitely many times in Σ(Ω). We will use the concept of cut-and-project sets for d = 2 and e = 1. With a slight abuse of notation, we will consider Ψ : R3 → C ' R2 . Then it is straightforward that for a cubic complex number γ, the set defined by (4.1) Σγ (Ω) = z ∈ Z[γ] : z 0 ∈ Ω , where Ω ⊆ R is an interval, is a cut-and-project set. Really, we have Ψγ (v0 , v1 , v2 ) = v0 + v1 γ + v2 γ 2 '
0 such that B(x, d/2) ⊆ T (x); • ∆(T (x)) is the minimal diameter D > 0 such that T (x) ⊆ B(x, D/2). These δ and ∆ allow us to compute the values of `m (γ) and Lm (γ), namely `m (γ) = inf δ T (x) and Lm (γ) = sup ∆ T (x) , x
x
where x runs the whole set X m (γ). A protocell of a point x is the set T (x) − x. We can define δ, ∆ analogously for the protocells. The set of all protocells of the tessellation of Σ(Ω) is called palette of Σ(Ω) and is denoted Pal(Ω). We therefore obtain that (5.2)
`m (γ) =
inf
T ∈Pal(Ω)
δ(T ) and Lm (γ) =
sup
∆(T ).
T ∈Pal(Ω)
For computing δ(T ) and ∆(T ), we will modify the approach of [MPZ03a], where 2-dimensional cut-and-project sets based on quadratic irrationalities are concerned. In the rest of this section, we consider γ satisfying the hypothesis of Theorem 4.1 m and Ω = [0, c) with c > 0 (however, not necessarily of the form c = 1−γ 0 ). Cut-and-project sets have finite local complexity. This implies that there are only finitely many protocells, i.e., the palette is finite. For any y ∈ Σ(Ω), the local configuration of size L around y is (5.3)
Σ(Ω) ∩ B(y, L) = y + Σ(Ω − y 0 ) ∩ B(0, L).
Therefore, there exists L > 0 such that (5.4) Σ(Ω − y10 ) ∩ B(0, L) = Σ(Ω − y20 ) ∩ B(0, L) =⇒ T (y1 ) − y1 = T (y2 ) − y2 , i.e., the protocells of y1 and y2 are identical when their neighborhoods of size L are identical. We give a way how to find such L, based on the following Lemma:
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T. HEJDA AND E. PELANTOVÁ
x3
x2
U 0 V
L/2
x1 Figure 1. To the proof of Lemma 5.1. Lemma 5.1. Let Ω = [0, c) be an interval. Let p be minimal positive integer such that =(γ p ) and =(γ) have the opposite signs. Let k be minimal integer such that (γ 0 )k ≤ c/2. Then γ i+j (γ i − γ j ) k (5.5) ∆ T (y) ≤ Lc := |γ| max for all y ∈ Σ(Ω). i,j∈{0,p−1,p} =(γ i γ j ) i<j
Proof. We first prove the statement for y = 0. The choice of p and k guarantees that x1 := γ k , x2 := γ k+p−1 and x3 := γ k+p satisfy x1 , x2 , x3 ∈ Σ(Ω), and that 0 is an inner point of triangle U with vertices x1 , x2 , x3 (see Figure 1. According to (5.1) we have V := {z ∈ C : |z − 0| ≤ |z − xj | for j = 1, 2, 3} ⊇ T (0). Let ρ be a radius of the smallest ball centered at 0 and containing the whole triangle V . From the definition of T (x) and ∆(T (x)) we see that ∆(T (0)) ≤ 2ρ. We can compute that the distances of vertices of V from the origin are given by 1 xi xj (xi − xj ) for i, j = 1, 2, 3 and i 6= j. 2 =(xi xj ) Thus the estimate (5.5) is valid for y = 0. It remains to show that it is valid for all y ∈ Σ(Ω). If y 0 ∈ [0, c/2) then y + xj for j = 1, 2, 3 are in Σ(Ω). If y 0 ∈ [c/2, c) then y − xj for j = 1, 2, 3 are in Σ(Ω). Both of these cases follow from the fact that x01 , x02 , x03 ∈ [0, c/2]. Therefore either x1 , x2 , x3 or −x1 , −x2 , −x3 are elements of Σ(Ω) − y, which means that the same estimate (5.5) can be used for any y ∈ Σ(Ω). Since Σ(Ω) is repetitive in our case, we have that `m (γ) = δ(T (x)) for infinitely many x ∈ X m (γ), and Lm (γ) = ∆(T (x)) for infinitely many x ∈ X m (γ). The algorithm to compute all protocells of the set Σ(Ω) for Ω = [0, c) is based on the following claim about them. Lemma 5.2. Let Ω = [0, c) be an interval. Then there exists a finite set Ξ = {ξ0 = 0 < ξ1 < · · · < ξN −1 < ξN = c} ⊂ [0, c] such that the mapping y 0 7→ T (y) − y is constant on [ξj−1 , ξj ) ∩ Z[γ 0 ] for each j = 1, . . . , N .
SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS
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Proof. Let us denote (5.6) Ξ := z 0 : z 0 ∈ Z[γ 0 ] ∩ [0, c) and |z| ≤ L ∪ c − z 0 : z 0 ∈ Z[γ 0 ] ∩ [0, c) and |z| ≤ L , where L := Lc is given by (5.5). The set Ξ is finite since it corresponds to a set of lattice points from Z3 such that their projections by both Ψ and Φ are bounded. Let x, y ∈ Σ(Ω) be such that x0 and y 0 are not seperated by a point from Ξ. We will show by contradiction that T (x) − x = T (y) − y. Assume the contrary. Then Σ(Ω − x0 ) ∩ B(0, L) 6= Σ(Ω − y 0 ) ∩ B(0, L). Without the loss of generality, there exists z ∈ Z[γ] such that z 0 ∈ Ω − x0 = [−x0 , c − x0 ), |z| ≤ L and z 0 ∈ / Ω − y 0 = [−y 0 , c − y 0 ). In the case x0 < y 0 , it yields x0 < c−z 0 ≤ y 0 . In the case x0 > y 0 , it yields y 0 < z 0 ≤ 0 x . In either case, x0 and y 0 are seperated by a point from Ξ — contradiction. The lemma gives a good upper bound on the number of distinct protocells: # Pal(Ω) ≤ 2# Σ(Ω) ∩ B(0, L) . Moreover, it allows us to compute all the protocells of the Voronoi tessellation of Σ(Ω) for a fixed Ω = [0, c): Algorithm 5.3. • Input: γ satisfying (2.1), Ω = [0, c). • Output: The pallete of Σ(Ω). (1) Compute the set Ξ = {ξ0 = 0 < ξ1 < · · · < ξN −1 < ξN = c} given by (5.6), with L := Lc as defined in (5.5). (2) For each interval [ξj , ξj+1 ) compute the local configuration of size L as Σ([−ξj , c − ξj )) ∩ B(0, L). (3) Compute the corresponding protocells to each of these intervals. (4) Remove possible duplicates in the list of protocells. The self-similararity property (cf. Proposition 4.2) allows us, when we study Σ(Ω) simultaneously for all Ω = [0, c) with c > 0, to fix aribtrary c0 > 0 and consider only values of c such that γ 0 c0 ≤ c < c0 . We are able to show that the pallete changes with c in a well-described way: Lemma 5.4. Let us fix b0 , c0 ∈ R such that 0 < b0 < c0 . Then there exists a finite set Θ = {θ1 < θ1 < · · · < θN −1 } ⊆ (b0 , c0 ) such that the mapping c 7→ Pal [0, c) is constant on each of the intervals (θj−1 , θj ) for j = 1, . . . , N , where we put θ0 := b0 and θN := c0 . Proof. Consider L := Lb0 defined in (5.5), and put (5.7) Θ := (Π0 − Π0 ) ∩ (b0 , c0 ), where Π0 := x0 ∈ Z[γ 0 ] : x0 ∈ (−c0 , c0 ) and |x| < L . We also let Π1 := {x0 ∈ Z[γ 0 ] : x ∈ B(0, L)}; clearly Π0 = Π1 ∩ (−c0 , c0 ). We will show that if for c1 , c2 ∈ [b0 , c0 ) there exists a protocell in the pallete for c1 that is not in the pallete for c2 , then necessarily a point of Θ lies between c1 and c2 .
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T. HEJDA AND E. PELANTOVÁ
x0 ∈ 0, 1
x0 ∈ 1, 1 + γ 0
x0 ∈ 2 + γ 0 , 1 +
1 γ0
x0 ∈ 1 + γ 0 , γ10
x0 ∈ 1 +
1 1 γ0 , γ02
x0 ∈
x0 ∈
1
1 ,1 γ02
γ0 , 2
+
+ γ0
1 γ02
Figure 2. Voronoi protocells for X 2 (γ) = Σ(Ω), where Ω = [0, 2/(1 − γ 0 )) and γ = γT is the complex Tribonacci constant. Let us take such a protocell. It corresponds to a subset of Π1 of the form S := Π1 ∩ [−a, −a + c1 ) with a ∈ [0, c1 ). Let us take the maximal interval (A, B) and the minimal interval [C, D] such that S = Π1 ∩ (A, B) = Π1 ∩ [C, D]. Then A, B, C, D ∈ Π1 , we have D −C ≤ c1 ≤ B −A and either c2 ≤ D −C or c2 ≥ B −A. In the first case c2 ≤ D − C, certainly C, D ∈ Π0 therefore D − C ∈ Θ. In the second case c2 ≥ B − A, we have 0 ∈ [−a, −a + c1 ) ⊂ (A, B), whence A < 0 and B > 0. Then B − A ≤ c2 gives A ≥ −c2 > −c0 and B ≤ c2 < c0 , therefore A, B ∈ Π0 and B − A ∈ Θ. Let us apply the lemma in the case b0 := γ 0 c0 . It gives us all possible cut-points of the interval [γ 0 c0 , c0 ) into sub-intervals on which the palette is stable. However, unlike in Lemma 5.2, in this lemma we cannot in general include the cases c ∈ Θ into any of the surrounding intervals, and these cases have to be studied seperately. Therefore, we can find all the palettes by the following algorithm: Algorithm 5.5. • Input: γ satisfying (2.1), c0 > 0. • Output: All possible palettes Pal(Ω) of Σ(Ω) for Ω = [0, c) and γ 0 c0 ≤ c < c0 . (1) Compute the set Θ = {θ0 < θ1 < · · · < θN −1 } given by (5.7), with L := Lγ 0 c0 as defined in (5.5). (2) Using Algorithm 5.3, compute the palettes Pal(Ω) for all Ω = [0, c) with 0 θN −2 +θN −1 1 c = γ 0 c0 , γ c02+θ0 , θ0 , θ0 +θ , θN −1 , θN −12 +c0 . 2 , θ1 , . . . , 2 (3) Remove possible duplicates in the list of palettes. 6. Complex Tribonacci number exploited. Proof of Theorem 1.2 In this section, we will describe the details of the proposed workflow on an example — the complex Tribonacci base γ = γT . We aim at the proof of Theorem 1.2. We put β := γγ = 1/γ 0 in the sequel.
SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS
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Figure 3. Part of the Voronoi tessellation of X 2 (γ) = Σ(Ω), where Ω = [0, 2/(1 − γ 0 )) and γ = γT is the complex Tribonacci constant. The theorem will be proved by combining the self-similarity property in Proposition 4.2 with the following proposition: Proposition 6.1. Let Ω = [0, c) with c ∈ (β 2 , β 3 ), where γ is the complex Tribonacci constant and β := 1/γ 0 . Denote Σ := Σ(Ω). Then s p β2 − 1 (6.1) min δ T (x) = 1/β and max ∆ T (x) = 2 β . x∈Σ x∈Σ 3β 2 − 1 Proof. We put c0 := β 3 . Since |1 − γ|, contradiction. k −2 Case k ≤ −2: We have |γ 0 | < 1, therefore m|γ 0 | ≤ |γ 0 | < |1 − γ 0 |, contradiction. • We can use the self-similarity property: It is straightforward that δ(γ j T ) = j |γ| δ(T ) and similarly for ∆; this gives analogous result for the values of ` and L. Proposition 6.1 corresponds to the case k = 2 of the theorem, since we have k = 2 if and only if c := m/(1 − γ 0 ) > (1/γ 0 )2 = β 2 and c < (1/γ 0 )3 = β 3 .√For this k, the values in (1.4) and (6.1) coincide (we remark that |γ| = β and 1/γ 0 = β). Remark. Let us point out that for a real base β the characteristic Lm (β) given by (1.1) is not influenced by gaps xk+1 − xk occurring only in a bounded piece of the real line. Therefore in general the value Lm (γ) as we have defined for the complex number γ is not the precise analogy to Lm (β). Nevertheless, if the set X m (γ) is repetitive (i.e., any patch occurs infinitely many times) then omitting configurations in a bounded area of the plane plays no role. 7. Delone tessellation — dual to Voronoi tessellation From Voronoi tessellation we can construct its dual tessellation: Let Σ ⊆ C be a Delone set. Consider a planar graph in C whose vertices are elements of the set Σ and edges are line segments connecting x, y ∈ Σ if any only if x and y are neighbors, i.e., their Voronoi cells T (x) and T (y) share a side. This graph divides the complex plane into faces; these faces are called Delone tiles. The collection of Delone tiles is Delone tessellation of Σ. All vertices of a Delone tile lie on a circle; its center is a vertex of the Voronoi tessellation. This is illustrated in Figure 4, which shows a small part of the set X 2 (γ), where γ is the complex Tribonacci constant; the quadrilateral is inscribed
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T. HEJDA AND E. PELANTOVÁ
Figure 5. Delone tiles of the set X 2 (γ), where γ is the complex Tribonacci constant. in the circle, the white cross marks the center of the circle and it is a common vertex of 4 Voronoi cells. The minimal distance inf x∈Σ δ(T (x)) is equal to the shortest edge in the Delone tiling. On the other hand, we have that the longest edge in the Delone tiling is (in general) shorter than supx∈Σ ∆(T (x)). Therefore, for a point x ∈ Σ(Ω) we can define ∆∗ T (x) := max |x − y| : y is a neighbor of x in Σ and study the maximum over all points x ∈ Σ. We can apply this to the sets X m (γ). We define L∗m (γ) := sup ∆∗ T (x) x∈X m (γ)
if X (γ) is Delone, and = +∞ otherwise. When X m (γ) is a cut-andproject set, we know that it has a finite local complexity and therefore finitely many different Delone prototiles. In the case of the complex Tribonacci base, the Delone prototiles of X 2 (γ) are depictied in Figure 5. From Table 1 we get the following result: m
L∗m (γ)
Theorem 7.1. Let γ = γT ≈ −0.771 + 1.115i be the complex root of the polynomial k Y 3 + Y 2 + Y − 1. Let m ∈ N. Find a maximal k ∈ Z such that m ≥ (1 − γ 0 ) γ10 , where γ 0 is the real Galois conjugate of γ. Then we have 3−k
L∗m (γ) = |γ|
.
8. Comments and open problems This paper treats a family of cubic complex Pisot units γ — such ones that the real number 1/γ 0 is positive and satisfies Property (F). We use the concept of cut-and-project sets to study the properties of the sets X m (γ). However, there are other cases where it might be possible to use this concept: (1) We can consider a different perspective of the Tribonacci constant. Let γ be the complex root of Y 3 + Y 2 + Y − 1, and put β := 1/γ 0 . Both γ and −γ are complex Pisot units. It was shown by Vávra [Váv13] that the real Tribonacci constant β has the so-called Property (−F). Shortly speaking, all numbers from I ∩ −β 1 , β+1 ), have a finite expansion of the Z[−1/β] = I ∩ Z[β], where I := ( β+1 a1 a2 a3 form −β + β 2 + −β 3 + · · · with aj ∈ {0, 1}. From this, we can show that X m (−γ) is a cut-and-project set for arbitrary m ≥ 1. The idea goes along the lines of the proof of Theorem 4.1. √ (2) Consider any real Pisot unit β of degree n. Let γ = i β. Then γ is a complex Pisot unit of degree at most 2n, its Galois conjugates are γ and √ ±i β 0 for β 0 conjugates of β.
SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS
15
√ Clearly X m (γ) = X m (−β)+i βX m (−β). Therefore the Voronoi cells of X m (γ) are rectangles. Values `m (γ) and Lm (γ) can be easily obtained from the minimal and maximal distances in X m (−β). In the case n = 2, relations between X m (−β) and cut-and-project sets in dimensions d = e = 1 were established in [MPP13], implying that X m (γ) is related to cut-and-project sets in dimensions d = e = 2. √ Let us note that Zaïmi [Zaï04] evaluated `m (γ) for γ = i β, m = bβ 2 c and β > 1 the root of Y 2 − aY − a, a ∈ N. (3) In the cubic case, we can weaken the hypothesis of Theorem 4.1. For a fixed m, the Property (F) can be replaced by the assumption that all numbers from Z[β] ∩ [0, 1) have a finite β-representation over the alphabet {0, 1, . . . , m}, where we denote β := 1/γ 0 > 1. Under such assumption, X m (γ) is a cut-and-project set. Akiyama, Rao and Steiner [ARS04] described precisely the set of purely periodic expansions of points from Z[β]. They have shown that all of them are of the form .ccc · · · = .cω , where 0 ≤ c < bβc and (a + b) | c. Since all numbers from Z[γ]∩[0, 1) have finite or periodic β-expansions (and the only periods are therefore the ones mentioned above), it is satisfactory to find m1 such that the number .(a + b)ω has a finite representation over the alphabet {0, . . . , m1 }. Under this hypothesis, all numbers from Z[γ] ∩ [0, 1) have a β finite representation over the alphabet {0, . . . , m} for all m ≥ m1 b a+b c. We were not able to establish the hypothesis in all cases. We list some cases in Table 2. (4) Quartic Pisot units γ with |γ| ∈ (1, 2) are treated by Dombek, Masáková and Ziegler in [DMZ13]. The authors study the question if every element of the ring Z[γ] of integers of Q(γ) can be written as a sum of distinct units. If the only units on the unit circle are ±1, then the question can be interpreted as Property (F) over the alphabet {−1, 0, 1}. Therefore the concept of cut-and-project sets can be applied to these quartic bases and symmetric alphabets as well. Let us conclude with several open questions. (A) Is it true that all real cubic Pisot units β with a complex conjugate satisfy the following: There exists m ∈ N such that all number from Z[β] ∩ [0, 1) have finite β-representation over the alphabet {0, . . . , m}? (B) Which real cubic bases −β, other than minus the Tribonacci constant, satisfy Property (−F)? (C) It is well known that in the real case, X m (β) is a relatively dense set in R+ if and only if m > β − 1. Can we establish analogous result in the complex case? In particular, is X m (γ) relatively dense set in C for all m > γγ − 1? Can the complex modification of the Feng’s result [Fen13] be proved? Namely that `m (γ) = 0 if and only if m > γγ − 1 and γ is not a complex Pisot number?
Acknowledgements We would like to thank Wolfgang Steiner for our fruitful discussions.
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T. HEJDA AND E. PELANTOVÁ
Representation of .(a + b)ω
b
a
m1
−2
≥3
2a − 2
.(a − 3)(2a − 2)(a − 3)(0)(1)
−3
≥7
3a − 6
.(a − 4)(2a − 5)(3a − 6)(a − 7)(0)(1)
=6
10
.(2)(7)(10)(10)(0)(0)(1)
=5
9
.(0)(9)(9)(5)(0)(0)(1)
=4
7
.(0)(2)(6)(7)(0)3 (1)
−4
≥8
8a − 11 .(a − 5)(2a − 11)(8a − 11)(4a − 31)(a − 8)(0)(1)
=7
39
.(0)(16)(39)(27)(0)3 (1)
=6
47
.(0)(3)(44)(47)(0)4 (1)
Table 2. List of pairs of a, b such that X m (γ) is a cut-and-project set, where γ is the non-real root of Y 3 + bY 2 + aY − 1 and m ≥ 0 m1 b 1/γ a+b c.
This work was supported by Grant Agency of the Czech Technical University in Prague grant SGS11/162/OHK4/3T/14, Czech Science Foundation grant 1303538S, and ANR/FWF project “FAN – Fractals and Numeration” (ANR-12-IS010002, FWF grant I1136). References [Aki00]
[ARS04]
[BH02] [BH03] [Bug96] [DMZ13] [EJK90]
[EJK98] [Fen13] [FW02] [KLP00]
[Kom02]
Shigeki Akiyama, Cubic Pisot units with finite beta expansions, Algebraic number theory and Diophantine analysis (Graz, 1998), de Gruyter, Berlin, 2000, pp. 11–26. MR 1770451 (2001i:11095) Shigeki Akiyama, Hui Rao, and Wolfgang Steiner, A certain finiteness property of Pisot number systems, J. Number Theory 107 (2004), no. 1, 135–160. MR 2059954 (2005g:11135) Peter Borwein and Kevin G. Hare, Some computations on the spectra of Pisot and Salem numbers, Math. Comp. 71 (2002), no. 238, 767–780. MR 1885627 (2003a:11135) , General forms for minimal spectral values for a class of quadratic Pisot numbers, Bull. London Math. Soc. 35 (2003), no. 1, 47–54. MR 1934431 (2003i:11154) Yann Bugeaud, On a property of Pisot numbers and related questions, Acta Math. Hungar. 73 (1996), no. 1-2, 33–39. MR 1415918 (98c:11113) Daniel Dombek, Zuzana Masáková, and Volker Ziegler, On distinct unit generated fields that are totally complex, 2013, submitted, 14 pp. Paul Erdős, Joó, and Vilmos Komornik, Characterization of the unique expanPIstván −ni and related problems, Bull. Soc. Math. France 118 (1990), no. 3, sions 1 = ∞ i=1 q 377–390. MR 1078082 (91j:11006) , On the sequence of numbers of the form 0 + 1 q + · · · + n q n , i ∈ {0, 1}, Acta Arith. 83 (1998), no. 3, 201–210. MR 1611185 (99a:11022) De-Jun Feng, On the topology of polynomials with bounded integer coefficients, 2013, preprint. De-Jun Feng and Zhi-Ying Wen, A property of Pisot numbers, J. Number Theory 97 (2002), no. 2, 305–316. MR 1942963 (2003i:11155) Vilmos Komornik, Paola Loreti, and Marco Pedicini, An approximation property of Pisot numbers, J. Number Theory 80 (2000), no. 2, 218–237. MR 1740512 (2000k:11116) Takao Komatsu, An approximation property of quadratic irrationals, Bull. Soc. Math. France 130 (2002), no. 1, 35–48. MR 1906191 (2003b:11063)
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[Moo97]
Robert V. Moody, Meyer sets and their duals, The mathematics of long-range aperiodic order (Waterloo, ON, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 489, Kluwer Acad. Publ., Dordrecht, 1997, pp. 403–441. MR 1460032 (98e:52029) [MPP13] Zuzana Masáková, Kateřina Pastirčáková, and Edita Pelantová, 2013, preprint. [MPZ03a] Zuzana Masáková, Jiří Patera, and Jan Zich, Classification of Voronoi and Delone tiles in quasicrystals. I. General method, J. Phys. A 36 (2003), no. 7, 1869–1894. MR 1960699 (2004a:52041) [MPZ03b] , Classification of Voronoi and Delone tiles of quasicrystals. II. Circular acceptance window of arbitrary size, J. Phys. A 36 (2003), no. 7, 1895–1912. MR 1960700 (2004a:52042) [MPZ05] , Classification of Voronoi and Delone tiles of quasicrystals. III. Decagonal acceptance window of any size, J. Phys. A 38 (2005), no. 9, 1947–1960. MR 2124374 (2005j:52025) [Sage] The Sage Group, Sage: Open source mathematical software (version 5.10), 2013, http: //www.sagemath.org. [Tik Z] Till Tantau et al., TikZ & PGF (version 2.10), 2010, http://sourceforge.net/ projects/pgf. [Váv13] Tomáš Vávra, Arithmetical aspects of a number system with negative Tribonacci base, Doktorandské dny (Praha, 2013), Czech Technical University in Prague, 2013, http: //kmwww.fjfi.cvut.cz/ddny/?loc=historie, pp. 301–308. [Zaï04] Toufik Zaïmi, On an approximation property of Pisot numbers. II, J. Théor. Nombres Bordeaux 16 (2004), no. 1, 239–249. MR 2145586 (2006f:11133) Dept. of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, Prague 12000, Czech Rep. Current address: LIAFA, CNRS UMR 7089, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France E-mail address: [email protected] Dept. of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, Prague 12000, Czech Rep. E-mail address: [email protected]
arXiv:1312.0653v1 [math.MG] 2 Dec 2013
SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS TOMÁŠ HEJDA AND EDITA PELANTOVÁ Abstract. For q ∈ R, q > 1, Erdős, Joó and Komornik study distances of the consecutive points in the set n nX o X m (q) = aj q j : n ∈ N, ak ∈ {0, 1, . . . , m} . j=0
The Pisot numbers play a crucial role for properties of X m (q). We follow work of Zaïmi who consideres X m (γ) with γ ∈ C \ R and γγ > 1. We show that for any non-real γ and m < γγ − 1, the set X m (γ) is not relatively dense in the complex plane. For a class of cubic complex Pisot units γ and m > γγ − 1 we deduce that X m (γ) is uniformly discrete and relatively dense, i.e., X m (γ) is a Delone set. For γ the complex root of Y 3 + Y 2 + Y − 1 we determine two parameters of the Delone set X m (γ) which are analogous to minimal and maximal distances for the real case X m (q).
1. Introduction In [EJK90, EJK98], Erdős, Joó and Komornik studied the set n nX o X m (β) := aj β j : n ∈ N, ak ∈ {0, 1, . . . , m} , j=0
where β > 1. Since this set has no accumulation points, we can find an increasing sequence 0 = x0 < x1 < x2 < · · · < xk < · · · such that X m (β) = {xk : k ∈ N}. The research of Erdős et al. aims to describe distances between consecutive points of X m (β), i.e., the sequence (xk+1 − xk )k∈N . The properties of this sequence depend on the value m ∈ N. It is easy to show that when m ≥ β − 1, we have xk+1 − xk ≤ 1 for all k ≥ 0; and when m < β − 1, the distances xk+1 − xk can be arbitrarily large. The properties of X m (β) depend on β being a Pisot number (i.e., an algebraic integer > 1 such that all its Galois conjugates are in modulus < 1). Bugeaud [Bug96] showed that `m (β) := lim inf (xk+1 − xk ) > 0 for all m ∈ N k→∞
if and only if the base β is a Pisot number. Recently, Feng [Fen13] proved a stronger result that the bound β − 1 for the alphabet size is crucial. In particular, `m (β) = 0 if and only if m > β − 1 and β is not a Pisot number. Therefore, the case β Pisot and m > β − 1 has been further studied. From the approximation property of Pisot numbers we know that for a fixed β and m > β − 1 2010 Mathematics Subject Classification. Primary 11A63, 52C23, 52C10; Secondary 11H99, 11-04.
1
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T. HEJDA AND E. PELANTOVÁ
the sequence (xk+1 − xk ) takes only finitely many values. Feng and Wen [FW02] used this fact to show that the sequence of distances (xk+1 − xk ) is substitutive, roughly speaking, can be generated by a system of rewriting rules over a finite alphabet. This allows, for a fixed β and m, to determine values of all distances (xk+1 − xk ) and subsequently the value of `m (β). An algorithm for obtaining the minimal distance `m (β) for certain β was as well proposed by Borwein and Hare [BH02]. The first formula which determines the value of `m (β) for all m at once appeared in 2000: Komornik, Loreti and Pedicini [KLP00] studied the base golden mean. The generalization of this result to all quadratic Pisot units was provided by Takao Komatsu [Kom02] in 2002. To the best of our knowledge, the value of (1.1)
Lm (β) := lim sup(xk+1 − xk ) k→∞
for all m is only known for the base Golden mean, due to Borwein and Hare [BH03]. Of course, for a given m, the value of Lm (β) can be computed using [FW02]. Zaïmi [Zaï04] was interested in a complementary question: Fix the alphabet size, i.e., the maximal digit m, and look for the extremal values of `m (β) where β runs through the Pisot numbers in (m, m + 1). Zaïmi showed that `m (β) is maximized for certain quadratic Pisot numbers. Besides that, Zaïmi started to study the set X m (γ) where he considered γ a complex number of modulus > 1, and he put (1.2) `m (γ) := inf |x − y| : x, y ∈ X m (γ), x 6= y . He proved an analogous result to the one for real bases by Bugeaud, namely that `m (γ) > 0 for all m if and only if γ is a complex Pisot number, which is defined as a non-real algebraic integer of modulus > 1 whose Galois conjugates except its complex conjugate are in modulus < 1. In the complex plane, `m (γ) and Lm (γ) cannot be defined as simply as in the real case since we have no natural ordering of the set X m (γ) in C. To overcome this, we will inspire by notions used in the definition of Delone sets. We say that a set Σ is: • uniformly discrete if there exists d > 0 such that |x − y| ≥ d for all distinct x, y ∈ Σ; • relatively dense if there exists D > 0 such that every ball B(x, D/2) of radius D/2 contains a point from Σ. A set that is both uniformly discrete and relatively dense is called Delone set. Clearly, if `m (γ) as given by (1.2) is positive, then X m (γ) is uniformly discrete and `m (γ) is the maximal d in the definition of uniform discreteness. Let us define Lm (γ) := inf D > 0 : B(x, D/2) ∩ X m (γ) 6= ∅ for all x ∈ C . In particular, Lm (γ) = +∞ if and only if X m (γ) is not relatively dense. The question for which pairs (γ, m) the set X m (γ) is uniformly discrete, and for which (γ, m) it is relatively dense is far from being solved. We provide a necessary condition for relative denseness and we show that in certain cases, it is sufficient as well: Theorem 1.1. Let γ ∈ C be a non-real number in modulus > 1.
SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS
3
(1) If m ≤ γγ − 1, then X m (γ) is not relatively dense. (2) [Zaï04] If m > γγ − 1 and γ is not an algebraic number, then X m (γ) is not uniformly discrete. The aim of this article is to study the sets X m (γ) simultaneously for all m ∈ N, for a certain class of cubic complex Pisot numbers with a positive conjugate γ 0 . For such γ the Rényi expansions in the base β := 1/γ 0 have nice properties which will be crucial in the proofs. When the base β = 1/γ 0 satisfies so-called Property (F), we show that for all sufficiently large m the set X m (γ) ⊆ C is a cut-and-project set; roughly speaking, X m (γ) is formed by projections of points from the lattice Z3 which lie in a sector bounded by two parallel planes in R3 , see Theorem 4.1. From that easily follows the asymptotic behaviour of `m (γ) and Lm (γ), namely: √ √ (1.3) `m (γ) = Θ(1/ m) and Lm (γ) = Θ(1/ m), √ √ √ where f (m) = Θ(1/ m) means that K1 / m ≤ f (m) ≤ K2 / m for some positive constants K1,2 . Any cut-and-project set Σ has finite local complexity, which means that there are only finitely many types of arrangements of close neighborhoods of points of Σ. More formally, for any % > 0 the set of patches (Σ − x) ∩ B(0, %) : x ∈ Σ is finite. In particular, it means that there are only finitely many Voronoi cells determined by the set Σ. The method of inspection of Voronoi cells for a specific cut-and-project set, as established by Masáková, Patera and Zich [MPZ03a, MPZ03b, MPZ05], enables us to give a general formula for both `m (γ) and Lm (γ). In the case that γ is the complex Tribonacci constant, i.e., the complex root of Y 3 + Y 2 + Y − 1, we get: Theorem 1.2. Let γ = γT ≈ −0.771 + 1.115i be the complex root of the polynomial Y 3 + Y 2 + Y − 1 and m ∈ N. Let k ∈ Z be the maximal integer such that m ≥ k (1 − γ 0 ) γ10 , where γ 0 is the real Galois conjugate of γ. Then we have r 1 − (γ 0 )2 3−k −k (1.4) `m (γ) = |γ| and Lm (γ) = 2 |γ| . 3 − (γ 0 )2 The article is organized as follows. In Section 2, we recall certain notions from the theory of β-expansions. Section 3 provides the proof of the 1st part of Theorem 1.1. In Section 4 we prove that X m (γ) is a cut-and-project set in certain cases. Section 5 describes the algorithms for computing `m (γ) and Lm (γ). These algorithms are applied to the complex Tribonacci number in Section 6, providing the proof of Theorem 1.2. In Section 7 we compute another characteristic of X m (γ) that is based on Delone tilings. Comments and open problems are in Section 8. All computations were carried out in C++ and in Sage [Sage]. The pictures were drawn using Tik Z [Tik Z]. 2. Preliminaries Let us recall some facts concerning β-expansions. For a real base β > 1, and for a number x ≥ 0, there exist unique N ∈ Z and unique integer coefficients
4
T. HEJDA AND E. PELANTOVÁ
aN , aN −1 , aN −2 , . . . such that aN 6= 0 and 0≤x−
N X
aj β j < β n
for all n ≤ N .
j=n
The string aN aN −1 · · · a1 a0 .a−1 a−2 · · · is called Rényi expansion of x in the base β. We immediatelly see that aj ∈ {0, . . . , bβc}. If only finitely many aj’s are nonzero, we speak about finite Rényi expansion of x. The set of numbers x ∈ R such that |x| has finite Rényi expansion is denoted Fin(β). We say that β > 1 satisfies Property (F) if Fin(β) is an algebraic ring, i.e., Fin(β) = Z[1/β], where Z[y] denotes as usual the integer combinations of powers of y. We will widely use the algebraic properties of cubic complex Pisot numbers γ. Such γ has two Galois conjugates. One of them is the complex conjugate γ. The second one is real and of modulus < 1, we will denote it γ 0 ; we have either −1 < γ 0 < 0 or 0 < γ 0 < 1. In general, for z ∈ Q(γ) we denote by z 0 ∈ Q(γ 0 ) ⊂ R its image under the Galois isomorphism that maps γ 7→ γ 0 . When γ is a unit (i.e., the absolute term of its minimal polynomial is ±1), we know that Z[1/γ] = Z[γ] = γZ[γ]. Akiyama [Aki00] described the real cubic units having Property (F) in terms of the coefficients of the minimal polynomial. Combining his result and Cardano’s formula we get that non-real γ is a cubic complex Pisot unit such that its real conjugate satisfies γ 0 > 0 and β := 1/γ 0 has Property (F) if and only if γ is a root of Y 3 + bY 2 + aY − 1, where a, b ∈ Z satisfy (2.1)
− 1 ≤ b ≤ a + 1,
18ab + 4a3 − a2 b2 − 4b3 + 27 > 0,
(a, b) 6= (1, −1).
In particular, the complex Tribonacci constant γT ≈ −0.771 + 1.115i (the root of Y 3 + Y 2 + Y − 1) falls into this scheme, and more generally, the complex roots of polynomials Y 3 + bY 2 + aY − 1 for b = 0, ±1 and a ≥ 1, with the exception (a, b) = (1, −1). 3. Proof of Theorem 1.1 We prove the first part of Theorem 1.1. We cannot easily follow the lines of the proof of the result for the real case (i.e., that m < β − 1 implies Lm (β) = +∞). In the proof of the theorem, the following ‘folklore’ lemma about the asymptotic density of relatively dense sets will be used: Lemma 3.1. Let Σ ⊂ C be a set without accumulation points. Suppose Σ is relatively dense. Then # Σ ∩ B(0, r) > 0. (3.1) lim inf r→∞ r2 Proof. Since Σ is relatively dense, there exists λ > 0 such that every square in C with side λ contains a point of Σ. Therefore every cell of the lattice λZ[i] = 2 {λa + iλb : a,√ b ∈ Z} contains a point of Σ. Since B(0, r) contains at least n cells, where n = r 2/λ , we get √ 2 # Σ ∩ B(0, r) r 2/λ 2 lim inf ≥ lim inf = 2 > 0. r→∞ r→∞ r2 r2 λ
SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS
5
Proof of Theorem 1.1, 1st statement. For simplicity, we denote Σ := X m (γ). First, we will show that for any r ≥ m we have Σ ∩ B 0, |γ|r − m ⊆ γ Σ ∩ B(0, r) + {0, . . . , m} and therefore (3.2)
# Σ ∩ B(0, |γ|r − m) ≤ (m + 1)# Σ ∩ B(0, r) . Pk To prove this, consider x = j=0 aj γ j with aj ∈ {0, . . . , m} and such that |x| ≤ Pk |γ|r − m. Then y := (x − a0 )/γ = j=1 aj γ j−1 ∈ Σ and |y| ≤ (|x| + a0 )/|γ| ≤ (|γ|r − m + m)/|γ| = r. Since x = γy + a0 , the inclusion is valid. Second, let us define recurrently r0 := m and rk+1 := |γ|rk − m. Clearly rk → ∞ whence rk+1 /rk → γ. Put nk := #(Σ ∩ B(0, rk ))/rk2 . Then (3.2) leads to nk+1 (m + 1)rk2 k→∞ m + 1 ≤ −−−−→ 2 < 1. 2 nk rk+1 |γ| This implies limk→∞ nk = 0, and subsequently lim inf r→∞ #(Σ ∩ B(0, r))/r2 = 0. Therefore the set Σ = X m (γ) is not relatively dense by Lemma 3.1. 4. Cut-and-project sets versus X m (γ) A cut-and-project scheme in dimension d + e consists of two linear maps Ψ : Rd+e → Rd and Φ : Rd+e → Re satisfying: (1) Ψ(Rd+e ) = Rd and restriction of Ψ to the lattice Zd+e is injective; (2) the set Φ(Zd+e ) is dense in Re . Let Ω ⊂ Re be a nonempty bounded set such that its closure equals the closure of its interior, i.e., Ω = Ω◦ . Then the set Σ(Ω) := Ψ(v) : v ∈ Zd+e , Φ(v) ∈ Ω ⊆ Rd is called cut-and-project set with acceptance window Ω. Cut-and-project sets can be defined in a slightly more general way, c.f. [Moo97]. It is well known that Σ(Ω) is a Delone set with finite local complexity. Moreover, in case e = 1, the form of acceptance window Ω = [l, r) or Ω = (l, r] guarantees that Σ(Ω) is repetitive, i.e., for every x ∈ Σ(Ω) and % > 0 the patch (Σ(Ω) − x) ∩ B(0, %) occurs infinitely many times in Σ(Ω). We will use the concept of cut-and-project sets for d = 2 and e = 1. With a slight abuse of notation, we will consider Ψ : R3 → C ' R2 . Then it is straightforward that for a cubic complex number γ, the set defined by (4.1) Σγ (Ω) = z ∈ Z[γ] : z 0 ∈ Ω , where Ω ⊆ R is an interval, is a cut-and-project set. Really, we have Ψγ (v0 , v1 , v2 ) = v0 + v1 γ + v2 γ 2 '
0 such that B(x, d/2) ⊆ T (x); • ∆(T (x)) is the minimal diameter D > 0 such that T (x) ⊆ B(x, D/2). These δ and ∆ allow us to compute the values of `m (γ) and Lm (γ), namely `m (γ) = inf δ T (x) and Lm (γ) = sup ∆ T (x) , x
x
where x runs the whole set X m (γ). A protocell of a point x is the set T (x) − x. We can define δ, ∆ analogously for the protocells. The set of all protocells of the tessellation of Σ(Ω) is called palette of Σ(Ω) and is denoted Pal(Ω). We therefore obtain that (5.2)
`m (γ) =
inf
T ∈Pal(Ω)
δ(T ) and Lm (γ) =
sup
∆(T ).
T ∈Pal(Ω)
For computing δ(T ) and ∆(T ), we will modify the approach of [MPZ03a], where 2-dimensional cut-and-project sets based on quadratic irrationalities are concerned. In the rest of this section, we consider γ satisfying the hypothesis of Theorem 4.1 m and Ω = [0, c) with c > 0 (however, not necessarily of the form c = 1−γ 0 ). Cut-and-project sets have finite local complexity. This implies that there are only finitely many protocells, i.e., the palette is finite. For any y ∈ Σ(Ω), the local configuration of size L around y is (5.3)
Σ(Ω) ∩ B(y, L) = y + Σ(Ω − y 0 ) ∩ B(0, L).
Therefore, there exists L > 0 such that (5.4) Σ(Ω − y10 ) ∩ B(0, L) = Σ(Ω − y20 ) ∩ B(0, L) =⇒ T (y1 ) − y1 = T (y2 ) − y2 , i.e., the protocells of y1 and y2 are identical when their neighborhoods of size L are identical. We give a way how to find such L, based on the following Lemma:
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T. HEJDA AND E. PELANTOVÁ
x3
x2
U 0 V
L/2
x1 Figure 1. To the proof of Lemma 5.1. Lemma 5.1. Let Ω = [0, c) be an interval. Let p be minimal positive integer such that =(γ p ) and =(γ) have the opposite signs. Let k be minimal integer such that (γ 0 )k ≤ c/2. Then γ i+j (γ i − γ j ) k (5.5) ∆ T (y) ≤ Lc := |γ| max for all y ∈ Σ(Ω). i,j∈{0,p−1,p} =(γ i γ j ) i<j
Proof. We first prove the statement for y = 0. The choice of p and k guarantees that x1 := γ k , x2 := γ k+p−1 and x3 := γ k+p satisfy x1 , x2 , x3 ∈ Σ(Ω), and that 0 is an inner point of triangle U with vertices x1 , x2 , x3 (see Figure 1. According to (5.1) we have V := {z ∈ C : |z − 0| ≤ |z − xj | for j = 1, 2, 3} ⊇ T (0). Let ρ be a radius of the smallest ball centered at 0 and containing the whole triangle V . From the definition of T (x) and ∆(T (x)) we see that ∆(T (0)) ≤ 2ρ. We can compute that the distances of vertices of V from the origin are given by 1 xi xj (xi − xj ) for i, j = 1, 2, 3 and i 6= j. 2 =(xi xj ) Thus the estimate (5.5) is valid for y = 0. It remains to show that it is valid for all y ∈ Σ(Ω). If y 0 ∈ [0, c/2) then y + xj for j = 1, 2, 3 are in Σ(Ω). If y 0 ∈ [c/2, c) then y − xj for j = 1, 2, 3 are in Σ(Ω). Both of these cases follow from the fact that x01 , x02 , x03 ∈ [0, c/2]. Therefore either x1 , x2 , x3 or −x1 , −x2 , −x3 are elements of Σ(Ω) − y, which means that the same estimate (5.5) can be used for any y ∈ Σ(Ω). Since Σ(Ω) is repetitive in our case, we have that `m (γ) = δ(T (x)) for infinitely many x ∈ X m (γ), and Lm (γ) = ∆(T (x)) for infinitely many x ∈ X m (γ). The algorithm to compute all protocells of the set Σ(Ω) for Ω = [0, c) is based on the following claim about them. Lemma 5.2. Let Ω = [0, c) be an interval. Then there exists a finite set Ξ = {ξ0 = 0 < ξ1 < · · · < ξN −1 < ξN = c} ⊂ [0, c] such that the mapping y 0 7→ T (y) − y is constant on [ξj−1 , ξj ) ∩ Z[γ 0 ] for each j = 1, . . . , N .
SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS
9
Proof. Let us denote (5.6) Ξ := z 0 : z 0 ∈ Z[γ 0 ] ∩ [0, c) and |z| ≤ L ∪ c − z 0 : z 0 ∈ Z[γ 0 ] ∩ [0, c) and |z| ≤ L , where L := Lc is given by (5.5). The set Ξ is finite since it corresponds to a set of lattice points from Z3 such that their projections by both Ψ and Φ are bounded. Let x, y ∈ Σ(Ω) be such that x0 and y 0 are not seperated by a point from Ξ. We will show by contradiction that T (x) − x = T (y) − y. Assume the contrary. Then Σ(Ω − x0 ) ∩ B(0, L) 6= Σ(Ω − y 0 ) ∩ B(0, L). Without the loss of generality, there exists z ∈ Z[γ] such that z 0 ∈ Ω − x0 = [−x0 , c − x0 ), |z| ≤ L and z 0 ∈ / Ω − y 0 = [−y 0 , c − y 0 ). In the case x0 < y 0 , it yields x0 < c−z 0 ≤ y 0 . In the case x0 > y 0 , it yields y 0 < z 0 ≤ 0 x . In either case, x0 and y 0 are seperated by a point from Ξ — contradiction. The lemma gives a good upper bound on the number of distinct protocells: # Pal(Ω) ≤ 2# Σ(Ω) ∩ B(0, L) . Moreover, it allows us to compute all the protocells of the Voronoi tessellation of Σ(Ω) for a fixed Ω = [0, c): Algorithm 5.3. • Input: γ satisfying (2.1), Ω = [0, c). • Output: The pallete of Σ(Ω). (1) Compute the set Ξ = {ξ0 = 0 < ξ1 < · · · < ξN −1 < ξN = c} given by (5.6), with L := Lc as defined in (5.5). (2) For each interval [ξj , ξj+1 ) compute the local configuration of size L as Σ([−ξj , c − ξj )) ∩ B(0, L). (3) Compute the corresponding protocells to each of these intervals. (4) Remove possible duplicates in the list of protocells. The self-similararity property (cf. Proposition 4.2) allows us, when we study Σ(Ω) simultaneously for all Ω = [0, c) with c > 0, to fix aribtrary c0 > 0 and consider only values of c such that γ 0 c0 ≤ c < c0 . We are able to show that the pallete changes with c in a well-described way: Lemma 5.4. Let us fix b0 , c0 ∈ R such that 0 < b0 < c0 . Then there exists a finite set Θ = {θ1 < θ1 < · · · < θN −1 } ⊆ (b0 , c0 ) such that the mapping c 7→ Pal [0, c) is constant on each of the intervals (θj−1 , θj ) for j = 1, . . . , N , where we put θ0 := b0 and θN := c0 . Proof. Consider L := Lb0 defined in (5.5), and put (5.7) Θ := (Π0 − Π0 ) ∩ (b0 , c0 ), where Π0 := x0 ∈ Z[γ 0 ] : x0 ∈ (−c0 , c0 ) and |x| < L . We also let Π1 := {x0 ∈ Z[γ 0 ] : x ∈ B(0, L)}; clearly Π0 = Π1 ∩ (−c0 , c0 ). We will show that if for c1 , c2 ∈ [b0 , c0 ) there exists a protocell in the pallete for c1 that is not in the pallete for c2 , then necessarily a point of Θ lies between c1 and c2 .
10
T. HEJDA AND E. PELANTOVÁ
x0 ∈ 0, 1
x0 ∈ 1, 1 + γ 0
x0 ∈ 2 + γ 0 , 1 +
1 γ0
x0 ∈ 1 + γ 0 , γ10
x0 ∈ 1 +
1 1 γ0 , γ02
x0 ∈
x0 ∈
1
1 ,1 γ02
γ0 , 2
+
+ γ0
1 γ02
Figure 2. Voronoi protocells for X 2 (γ) = Σ(Ω), where Ω = [0, 2/(1 − γ 0 )) and γ = γT is the complex Tribonacci constant. Let us take such a protocell. It corresponds to a subset of Π1 of the form S := Π1 ∩ [−a, −a + c1 ) with a ∈ [0, c1 ). Let us take the maximal interval (A, B) and the minimal interval [C, D] such that S = Π1 ∩ (A, B) = Π1 ∩ [C, D]. Then A, B, C, D ∈ Π1 , we have D −C ≤ c1 ≤ B −A and either c2 ≤ D −C or c2 ≥ B −A. In the first case c2 ≤ D − C, certainly C, D ∈ Π0 therefore D − C ∈ Θ. In the second case c2 ≥ B − A, we have 0 ∈ [−a, −a + c1 ) ⊂ (A, B), whence A < 0 and B > 0. Then B − A ≤ c2 gives A ≥ −c2 > −c0 and B ≤ c2 < c0 , therefore A, B ∈ Π0 and B − A ∈ Θ. Let us apply the lemma in the case b0 := γ 0 c0 . It gives us all possible cut-points of the interval [γ 0 c0 , c0 ) into sub-intervals on which the palette is stable. However, unlike in Lemma 5.2, in this lemma we cannot in general include the cases c ∈ Θ into any of the surrounding intervals, and these cases have to be studied seperately. Therefore, we can find all the palettes by the following algorithm: Algorithm 5.5. • Input: γ satisfying (2.1), c0 > 0. • Output: All possible palettes Pal(Ω) of Σ(Ω) for Ω = [0, c) and γ 0 c0 ≤ c < c0 . (1) Compute the set Θ = {θ0 < θ1 < · · · < θN −1 } given by (5.7), with L := Lγ 0 c0 as defined in (5.5). (2) Using Algorithm 5.3, compute the palettes Pal(Ω) for all Ω = [0, c) with 0 θN −2 +θN −1 1 c = γ 0 c0 , γ c02+θ0 , θ0 , θ0 +θ , θN −1 , θN −12 +c0 . 2 , θ1 , . . . , 2 (3) Remove possible duplicates in the list of palettes. 6. Complex Tribonacci number exploited. Proof of Theorem 1.2 In this section, we will describe the details of the proposed workflow on an example — the complex Tribonacci base γ = γT . We aim at the proof of Theorem 1.2. We put β := γγ = 1/γ 0 in the sequel.
SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS
11
Figure 3. Part of the Voronoi tessellation of X 2 (γ) = Σ(Ω), where Ω = [0, 2/(1 − γ 0 )) and γ = γT is the complex Tribonacci constant. The theorem will be proved by combining the self-similarity property in Proposition 4.2 with the following proposition: Proposition 6.1. Let Ω = [0, c) with c ∈ (β 2 , β 3 ), where γ is the complex Tribonacci constant and β := 1/γ 0 . Denote Σ := Σ(Ω). Then s p β2 − 1 (6.1) min δ T (x) = 1/β and max ∆ T (x) = 2 β . x∈Σ x∈Σ 3β 2 − 1 Proof. We put c0 := β 3 . Since |1 − γ|, contradiction. k −2 Case k ≤ −2: We have |γ 0 | < 1, therefore m|γ 0 | ≤ |γ 0 | < |1 − γ 0 |, contradiction. • We can use the self-similarity property: It is straightforward that δ(γ j T ) = j |γ| δ(T ) and similarly for ∆; this gives analogous result for the values of ` and L. Proposition 6.1 corresponds to the case k = 2 of the theorem, since we have k = 2 if and only if c := m/(1 − γ 0 ) > (1/γ 0 )2 = β 2 and c < (1/γ 0 )3 = β 3 .√For this k, the values in (1.4) and (6.1) coincide (we remark that |γ| = β and 1/γ 0 = β). Remark. Let us point out that for a real base β the characteristic Lm (β) given by (1.1) is not influenced by gaps xk+1 − xk occurring only in a bounded piece of the real line. Therefore in general the value Lm (γ) as we have defined for the complex number γ is not the precise analogy to Lm (β). Nevertheless, if the set X m (γ) is repetitive (i.e., any patch occurs infinitely many times) then omitting configurations in a bounded area of the plane plays no role. 7. Delone tessellation — dual to Voronoi tessellation From Voronoi tessellation we can construct its dual tessellation: Let Σ ⊆ C be a Delone set. Consider a planar graph in C whose vertices are elements of the set Σ and edges are line segments connecting x, y ∈ Σ if any only if x and y are neighbors, i.e., their Voronoi cells T (x) and T (y) share a side. This graph divides the complex plane into faces; these faces are called Delone tiles. The collection of Delone tiles is Delone tessellation of Σ. All vertices of a Delone tile lie on a circle; its center is a vertex of the Voronoi tessellation. This is illustrated in Figure 4, which shows a small part of the set X 2 (γ), where γ is the complex Tribonacci constant; the quadrilateral is inscribed
14
T. HEJDA AND E. PELANTOVÁ
Figure 5. Delone tiles of the set X 2 (γ), where γ is the complex Tribonacci constant. in the circle, the white cross marks the center of the circle and it is a common vertex of 4 Voronoi cells. The minimal distance inf x∈Σ δ(T (x)) is equal to the shortest edge in the Delone tiling. On the other hand, we have that the longest edge in the Delone tiling is (in general) shorter than supx∈Σ ∆(T (x)). Therefore, for a point x ∈ Σ(Ω) we can define ∆∗ T (x) := max |x − y| : y is a neighbor of x in Σ and study the maximum over all points x ∈ Σ. We can apply this to the sets X m (γ). We define L∗m (γ) := sup ∆∗ T (x) x∈X m (γ)
if X (γ) is Delone, and = +∞ otherwise. When X m (γ) is a cut-andproject set, we know that it has a finite local complexity and therefore finitely many different Delone prototiles. In the case of the complex Tribonacci base, the Delone prototiles of X 2 (γ) are depictied in Figure 5. From Table 1 we get the following result: m
L∗m (γ)
Theorem 7.1. Let γ = γT ≈ −0.771 + 1.115i be the complex root of the polynomial k Y 3 + Y 2 + Y − 1. Let m ∈ N. Find a maximal k ∈ Z such that m ≥ (1 − γ 0 ) γ10 , where γ 0 is the real Galois conjugate of γ. Then we have 3−k
L∗m (γ) = |γ|
.
8. Comments and open problems This paper treats a family of cubic complex Pisot units γ — such ones that the real number 1/γ 0 is positive and satisfies Property (F). We use the concept of cut-and-project sets to study the properties of the sets X m (γ). However, there are other cases where it might be possible to use this concept: (1) We can consider a different perspective of the Tribonacci constant. Let γ be the complex root of Y 3 + Y 2 + Y − 1, and put β := 1/γ 0 . Both γ and −γ are complex Pisot units. It was shown by Vávra [Váv13] that the real Tribonacci constant β has the so-called Property (−F). Shortly speaking, all numbers from I ∩ −β 1 , β+1 ), have a finite expansion of the Z[−1/β] = I ∩ Z[β], where I := ( β+1 a1 a2 a3 form −β + β 2 + −β 3 + · · · with aj ∈ {0, 1}. From this, we can show that X m (−γ) is a cut-and-project set for arbitrary m ≥ 1. The idea goes along the lines of the proof of Theorem 4.1. √ (2) Consider any real Pisot unit β of degree n. Let γ = i β. Then γ is a complex Pisot unit of degree at most 2n, its Galois conjugates are γ and √ ±i β 0 for β 0 conjugates of β.
SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS
15
√ Clearly X m (γ) = X m (−β)+i βX m (−β). Therefore the Voronoi cells of X m (γ) are rectangles. Values `m (γ) and Lm (γ) can be easily obtained from the minimal and maximal distances in X m (−β). In the case n = 2, relations between X m (−β) and cut-and-project sets in dimensions d = e = 1 were established in [MPP13], implying that X m (γ) is related to cut-and-project sets in dimensions d = e = 2. √ Let us note that Zaïmi [Zaï04] evaluated `m (γ) for γ = i β, m = bβ 2 c and β > 1 the root of Y 2 − aY − a, a ∈ N. (3) In the cubic case, we can weaken the hypothesis of Theorem 4.1. For a fixed m, the Property (F) can be replaced by the assumption that all numbers from Z[β] ∩ [0, 1) have a finite β-representation over the alphabet {0, 1, . . . , m}, where we denote β := 1/γ 0 > 1. Under such assumption, X m (γ) is a cut-and-project set. Akiyama, Rao and Steiner [ARS04] described precisely the set of purely periodic expansions of points from Z[β]. They have shown that all of them are of the form .ccc · · · = .cω , where 0 ≤ c < bβc and (a + b) | c. Since all numbers from Z[γ]∩[0, 1) have finite or periodic β-expansions (and the only periods are therefore the ones mentioned above), it is satisfactory to find m1 such that the number .(a + b)ω has a finite representation over the alphabet {0, . . . , m1 }. Under this hypothesis, all numbers from Z[γ] ∩ [0, 1) have a β finite representation over the alphabet {0, . . . , m} for all m ≥ m1 b a+b c. We were not able to establish the hypothesis in all cases. We list some cases in Table 2. (4) Quartic Pisot units γ with |γ| ∈ (1, 2) are treated by Dombek, Masáková and Ziegler in [DMZ13]. The authors study the question if every element of the ring Z[γ] of integers of Q(γ) can be written as a sum of distinct units. If the only units on the unit circle are ±1, then the question can be interpreted as Property (F) over the alphabet {−1, 0, 1}. Therefore the concept of cut-and-project sets can be applied to these quartic bases and symmetric alphabets as well. Let us conclude with several open questions. (A) Is it true that all real cubic Pisot units β with a complex conjugate satisfy the following: There exists m ∈ N such that all number from Z[β] ∩ [0, 1) have finite β-representation over the alphabet {0, . . . , m}? (B) Which real cubic bases −β, other than minus the Tribonacci constant, satisfy Property (−F)? (C) It is well known that in the real case, X m (β) is a relatively dense set in R+ if and only if m > β − 1. Can we establish analogous result in the complex case? In particular, is X m (γ) relatively dense set in C for all m > γγ − 1? Can the complex modification of the Feng’s result [Fen13] be proved? Namely that `m (γ) = 0 if and only if m > γγ − 1 and γ is not a complex Pisot number?
Acknowledgements We would like to thank Wolfgang Steiner for our fruitful discussions.
16
T. HEJDA AND E. PELANTOVÁ
Representation of .(a + b)ω
b
a
m1
−2
≥3
2a − 2
.(a − 3)(2a − 2)(a − 3)(0)(1)
−3
≥7
3a − 6
.(a − 4)(2a − 5)(3a − 6)(a − 7)(0)(1)
=6
10
.(2)(7)(10)(10)(0)(0)(1)
=5
9
.(0)(9)(9)(5)(0)(0)(1)
=4
7
.(0)(2)(6)(7)(0)3 (1)
−4
≥8
8a − 11 .(a − 5)(2a − 11)(8a − 11)(4a − 31)(a − 8)(0)(1)
=7
39
.(0)(16)(39)(27)(0)3 (1)
=6
47
.(0)(3)(44)(47)(0)4 (1)
Table 2. List of pairs of a, b such that X m (γ) is a cut-and-project set, where γ is the non-real root of Y 3 + bY 2 + aY − 1 and m ≥ 0 m1 b 1/γ a+b c.
This work was supported by Grant Agency of the Czech Technical University in Prague grant SGS11/162/OHK4/3T/14, Czech Science Foundation grant 1303538S, and ANR/FWF project “FAN – Fractals and Numeration” (ANR-12-IS010002, FWF grant I1136). References [Aki00]
[ARS04]
[BH02] [BH03] [Bug96] [DMZ13] [EJK90]
[EJK98] [Fen13] [FW02] [KLP00]
[Kom02]
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[Moo97]
Robert V. Moody, Meyer sets and their duals, The mathematics of long-range aperiodic order (Waterloo, ON, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 489, Kluwer Acad. Publ., Dordrecht, 1997, pp. 403–441. MR 1460032 (98e:52029) [MPP13] Zuzana Masáková, Kateřina Pastirčáková, and Edita Pelantová, 2013, preprint. [MPZ03a] Zuzana Masáková, Jiří Patera, and Jan Zich, Classification of Voronoi and Delone tiles in quasicrystals. I. General method, J. Phys. A 36 (2003), no. 7, 1869–1894. MR 1960699 (2004a:52041) [MPZ03b] , Classification of Voronoi and Delone tiles of quasicrystals. II. Circular acceptance window of arbitrary size, J. Phys. A 36 (2003), no. 7, 1895–1912. MR 1960700 (2004a:52042) [MPZ05] , Classification of Voronoi and Delone tiles of quasicrystals. III. Decagonal acceptance window of any size, J. Phys. A 38 (2005), no. 9, 1947–1960. MR 2124374 (2005j:52025) [Sage] The Sage Group, Sage: Open source mathematical software (version 5.10), 2013, http: //www.sagemath.org. [Tik Z] Till Tantau et al., TikZ & PGF (version 2.10), 2010, http://sourceforge.net/ projects/pgf. [Váv13] Tomáš Vávra, Arithmetical aspects of a number system with negative Tribonacci base, Doktorandské dny (Praha, 2013), Czech Technical University in Prague, 2013, http: //kmwww.fjfi.cvut.cz/ddny/?loc=historie, pp. 301–308. [Zaï04] Toufik Zaïmi, On an approximation property of Pisot numbers. II, J. Théor. Nombres Bordeaux 16 (2004), no. 1, 239–249. MR 2145586 (2006f:11133) Dept. of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, Prague 12000, Czech Rep. Current address: LIAFA, CNRS UMR 7089, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France E-mail address: [email protected] Dept. of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, Prague 12000, Czech Rep. E-mail address: [email protected]
