SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS

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1 December 2, 203 SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS TOMÁŠ HEJDA AND EDITA PELANTOVÁ Abstract. For q R, q >, Erdős, Joó and Komornik study distances of the consecutive points in the set { n } X m (q) = a j q j : n N, a k {0,,..., 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 >. We show that for any non-real γ and m < γγ, 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 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 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).. Introduction In [EJK90, EJK98], Erdős, Joó and Komornik studied the set { n } X m () := a j j : n N, a k {0,,..., m}, j=0 where >. Since this set has no accumulation points, we can find an increasing sequence 0 = x 0 < x < x 2 < < x k < such that X m () = {x k : k N}. The research of Erdős et al. aims to describe distances between consecutive points of X m (), i.e., the sequence (x k+ x k ) k N. The properties of this sequence depend on the value m N. It is easy to show that when m, we have x k+ x k for all k 0; and when m <, the distances x k+ x k can be arbitrarily large. The properties of X m () depend on being a Pisot number (i.e., an algebraic integer > such that all its Galois conjugates are in modulus < ). Bugeaud [Bug96] showed that l m () := lim inf (x k+ x k ) > 0 for all m N k if and only if the base is a Pisot number. Recently, Feng [Fen3] proved a stronger result that the bound for the alphabet size is crucial. In particular, l m () = 0 if and only if m > and is not a Pisot number. Therefore, the case Pisot and m > has been further studied. From the approximation property of Pisot numbers we know that for a fixed and m > 200 Mathematics Subject Classification. Primary A63, 52C23, 52C0; Secondary H99, -04.

2 2 T. HEJDA AND E. PELANTOVÁ the sequence (x k+ x k ) takes only finitely many values. Feng and Wen [FW02] used this fact to show that the sequence of distances (x k+ x k ) is substitutive, roughly speaking, can be generated by a system of rewriting rules over a finite alphabet. This allows, for a fixed and m, to determine values of all distances (x k+ x k ) and subsequently the value of l m (). An algorithm for obtaining the minimal distance l m () for certain was as well proposed by Borwein and Hare [BH02]. The first formula which determines the value of l m () 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 To the best of our knowledge, the value of (.) L m () := lim sup(x k+ x k ) 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 L m () 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 l m () where runs through the Pisot numbers in (m, m + ). Zaïmi showed that l m () 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 >, and he put (.2) l m (γ) := inf { x y : x, y X m (γ), x y }. He proved an analogous result to the one for real bases by Bugeaud, namely that l 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 > whose Galois conjugates except its complex conjugate are in modulus <. In the complex plane, l m (γ) and L m (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 l m (γ) as given by (.2) is positive, then X m (γ) is uniformly discrete and l m (γ) is the maximal d in the definition of uniform discreteness. Let us define L m (γ) := inf { D > 0 : B(x, D/2) X m (γ) for all x C }. In particular, L m (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.. Let γ C be a non-real number in modulus >.

3 SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS 3 () If m γγ, then X m (γ) is not relatively dense. (2) [Zaï04] If m > γγ 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. For such γ the Rényi expansions in the base := /γ have nice properties which will be crucial in the proofs. When the base = /γ 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 Z 3 which lie in a sector bounded by two parallel planes in R 3, see Theorem 4.. From that easily follows the asymptotic behaviour of l m (γ) and L m (γ), namely: (.3) l m (γ) = Θ(/ m) and L m (γ) = Θ(/ m), where f(m) = Θ(/ m) means that K / m f(m) K 2 / m for some positive constants K,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 ϱ > 0 the set of patches { (Σ x) B(0, ϱ) : 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 l m (γ) and L m (γ). In the case that γ is the complex Tribonacci constant, i.e., the complex root of Y 3 + Y 2 + Y, we get: Theorem.2. Let γ = γ T i be the complex root of the polynomial Y 3 + Y 2 + Y and m N. Let k Z be the maximal integer such that m ( γ ) ( ) k, γ where γ is the real Galois conjugate of γ. Then we have (.4) l m (γ) = γ k (γ ) 2 and L m (γ) = 2 3 (γ ) 2 γ 3 k. The article is organized as follows. In Section 2, we recall certain notions from the theory of -expansions. Section 3 provides the proof of the st part of Theorem.. 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 l m (γ) and L m (γ). These algorithms are applied to the complex Tribonacci number in Section 6, providing the proof of Theorem.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 TikZ [TikZ]. 2. Preliminaries Let us recall some facts concerning -expansions. For a real base >, and for a number x 0, there exist unique N Z and unique integer coefficients

4 4 T. HEJDA AND E. PELANTOVÁ a N, a N, a N 2,... such that a N 0 and N 0 x a j j < n for all n N. j=n The string a N a N a a 0.a a 2 is called Rényi expansion of x in the base. We immediatelly see that a j {0,..., }. If only finitely many a j 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(). We say that > satisfies Property (F) if Fin() is an algebraic ring, i.e., Fin() = Z[/], 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 <, we will denote it γ ; we have either < γ < 0 or 0 < γ <. In general, for z Q(γ) we denote by z Q(γ ) R its image under the Galois isomorphism that maps γ γ. When γ is a unit (i.e., the absolute term of its minimal polynomial is ±), we know that Z[/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 and := /γ has Property (F) if and only if γ is a root of Y 3 + by 2 + ay, where a, b Z satisfy (2.) b a +, 8ab + 4a 3 a 2 b 2 4b > 0, (a, b) (, ). In particular, the complex Tribonacci constant γ T i (the root of Y 3 + Y 2 + Y ) falls into this scheme, and more generally, the complex roots of polynomials Y 3 + by 2 + ay for b = 0, ± and a, with the exception (a, b) = (, ). 3. Proof of Theorem. We prove the first part of Theorem.. We cannot easily follow the lines of the proof of the result for the real case (i.e., that m < implies L m () = + ). In the proof of the theorem, the following folklore lemma about the asymptotic density of relatively dense sets will be used: Lemma 3.. Let Σ C be a set without accumulation points. relatively dense. Then # ( Σ B(0, r) ) (3.) lim inf r r 2 > 0. Suppose Σ is 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] = {λa + iλb : a, b Z} contains a point of Σ. Since B(0, r) contains at least n 2 cells, where n = r 2/λ, we get 2 r 2/λ lim inf r # ( Σ B(0, r) ) r 2 lim inf r r 2 = 2 λ 2 > 0.

5 SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS 5 Proof of Theorem., st statement. For simplicity, we denote Σ := X m (γ). First, we will show that for any r m we have and therefore Σ B ( 0, γ r m ) γ ( Σ B(0, r) ) + {0,..., m} (3.2) # ( Σ B(0, γ r m) ) (m + )# ( Σ B(0, r) ). To prove this, consider x = k j=0 a jγ j with a j {0,..., m} and such that x γ r m. Then y := (x a 0 )/γ = k j= a jγ j Σ and y ( x + a 0 )/ γ ( γ r m + m)/ γ = r. Since x = γy + a 0, the inclusion is valid. Second, let us define recurrently r 0 := m and r k+ := γ r k m. Clearly r k whence r k+ /r k γ. Put n k := #(Σ B(0, r k ))/rk 2. Then (3.2) leads to n k+ n k (m + )r2 k r 2 k+ k m + γ 2 <. This implies lim k n k = 0, and subsequently lim inf r #(Σ B(0, r))/r 2 = 0. Therefore the set Σ = X m (γ) is not relatively dense by Lemma Cut-and-project sets versus X m (γ) A cut-and-project scheme in dimension d + e consists of two linear maps Ψ : R d+e R d and Φ : R d+e R e satisfying: () Ψ(R d+e ) = R d and restriction of Ψ to the lattice Z d+e is injective; (2) the set Φ(Z d+e ) is dense in R e. Let Ω R e 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 Z d+e, Φ(v) Ω } R d 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 =, 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 ϱ > 0 the patch (Σ(Ω) x) B(0, ϱ) occurs infinitely many times in Σ(Ω). We will use the concept of cut-and-project sets for d = 2 and e =. With a slight abuse of notation, we will consider Ψ : R 3 C R 2. Then it is straightforward that for a cubic complex number γ, the set defined by (4.) Σ γ (Ω) = { z Z[γ] : z Ω }, where Ω R is an interval, is a cut-and-project set. Really, we have Ψ γ (v 0, v, v 2 ) = v 0 + v γ + v 2 γ 2 Φ γ (v 0, v, v 2 ) = v 0 + v γ + v 2 (γ ) 2. ( R(v0 + v γ + v 2 γ 2 ) ) I(v 0 + v γ + v 2 γ 2, ) We will often omit the index γ in the sequel. We now show how X m (γ) fit into the cut-and-project scheme:

6 6 T. HEJDA AND E. PELANTOVÁ Theorem 4.. Let γ be a cubic complex Pisot unit with a positive conjugate γ, and let m be an integer m γγ. Suppose that the base /γ has Property (F). Then X m (γ) is a cut-and-project set, namely (4.2) X m (γ) = Σ(Ω) = { z Z[γ] : z Ω } with Ω = [ 0, m/( γ ) ). Proof. Inclusion : Let z X m (γ). Then z = n j=0 a jγ j with a j {0,..., m} and clearly z Z[γ]. Moreover, 0 z = n a j (γ ) j k=0 n m(γ ) j < m γ. Inclusion : Let us take z Z[γ] with z Ω. Denote = /γ = γγ. We will discuss the following two cases: k=0 () Suppose 0 z <. The real base has Property (F) by the hypothesis, therefore every number from Z[/] [0, ) has a finite expansion 0.a a 2 a 3... a n over the alphabet {0,..., m 0 }, where m 0 := (the expansion certainly starts after the fractional point since z < ). This means that z X m0 (γ) X m (γ), we get z X m (γ). (2) Suppose z < m/( γ ). Since z < j=0 m j, there exists a = n j= a j j and therefore z = n j= a jγ j X m0 (γ). Since minimal k 0 such that z k j=0 m j < 0. Let b {0,..., m} be such that k 0 z m j b k < k j=0 (this is possible because m > ). Then u := k (z k ) m j b k j=0 satisfies 0 u <. Then by the previous case there exist a,..., a n {0,..., m 0 } such that u = n j= a j j. Altogehter, and z X m (γ). k z = m(γ ) j + b(γ ) k + j=0 k+n j=k+ a j k (γ ) j The property of cut-and-project sets, which allows us to determing the values of l m (γ) and L m (γ), is self-similarity. We say that a Delone set Σ C is self-similar with factor κ C if κσ Σ. In general, cut-and-project sets are not self-similar. In our special case (4.), not only that the sets are self-similar, we can prove even stronger property that will be useful later: Proposition 4.2. Let γ be a cubic complex Pisot unit. Then Σ ( (γ ) k Ω ) = γ k Σ(Ω) for any interval Ω and any k Z. The self-similarity for κ = γ and Ω = [0, c) follows from the fact that Σ(γ Ω) Σ(Ω).

7 SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS 7 Proof. We will prove the claim for k =, the general case follows by induction. Because Z[γ] = γz[γ], we have that Σ(γ Ω) = { x Z[γ] : x γ Ω } = { x Z[γ] : γ x Ω } = { γx γz[γ] : x Ω } = γσ(ω). Remark. Self-similarity of Σ(Ω) implies the asymptotic behaviour of l m (γ) and L m (γ), as described in (.3), because we have that γ = / γ. 5. Voronoi tessellations In a Delone set Σ, the Voronoi cell of a point x Σ is the set of points which are closer to x than to any other point in Σ, formally (5.) T (x) = { z C : z x z y for all y Σ }. The cell is a convex polygon having x as an interior point. Clearly x Σ T (x) = C and T (x) T (y) = for any distinct x, y Σ. This means that the collection of cells {T (x) : x Σ} is a tessellation of the complex plane. For every cell T (x) we define two characteristics: δ(t (x)) is the maximal diameter d > 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 l m (γ) and L m (γ), namely l m (γ) = inf δ( T (x) ) and L m (γ) = sup ( T (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) l m (γ) = inf δ(t ) and L m(γ) = sup (T ). T Pal(Ω) 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. and Ω = [0, c) with c > 0 (however, not necessarily of the form c = m γ ). 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 ) B(0, L). Therefore, there exists L > 0 such that (5.4) Σ(Ω y ) B(0, L) = Σ(Ω y 2) B(0, L) = T (y ) y = T (y 2 ) y 2, i.e., the protocells of y and y 2 are identical when their neighborhoods of size L are identical. We give a way how to find such L, based on the following Lemma: x

8 8 T. HEJDA AND E. PELANTOVÁ x 3 x 2 U V 0 L/2 x Figure. To the proof of Lemma 5.. Lemma 5.. Let Ω = [0, c) be an interval. Let p be minimal positive integer such that I(γ p ) and I(γ) have the opposite signs. Let k be minimal integer such that (γ ) k c/2. Then (5.5) ( T (y) ) L c := γ k max i,j {0,p,p} i<j γi+j (γ i γ j ) I(γ i γ j ) for all y Σ(Ω). Proof. We first prove the statement for y = 0. The choice of p and k guarantees that x := γ k, x 2 := γ k+p and x 3 := γ k+p satisfy x, x 2, x 3 Σ(Ω), and that 0 is an inner point of triangle U with vertices x, x 2, x 3 (see Figure. According to (5.) we have V := {z C : z 0 z x j for j =, 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 x ix j (x i x j ) for i, j =, 2, 3 and i j. 2 I(x i x j ) 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, c/2) then y + x j for j =, 2, 3 are in Σ(Ω). If y [c/2, c) then y x j for j =, 2, 3 are in Σ(Ω). Both of these cases follow from the fact that x, x 2, x 3 [0, c/2]. Therefore either x, x 2, x 3 or x, x 2, x 3 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 l m (γ) = δ(t (x)) for infinitely many x X m (γ), and L m (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 = {ξ 0 = 0 < ξ < < ξ N < ξ N = c} [0, c] such that the mapping y T (y) y is constant on [ξ j, ξ j ) Z[γ ] for each j =,..., N.

9 SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS 9 Proof. Let us denote (5.6) Ξ := { z : z Z[γ ] [0, c) and z L } { c z : z Z[γ ] [0, c) and z L }, where L := L c is given by (5.5). The set Ξ is finite since it corresponds to a set of lattice points from Z 3 such that their projections by both Ψ and Φ are bounded. Let x, y Σ(Ω) be such that x and y 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 x ) B(0, L) Σ(Ω y ) B(0, L). Without the loss of generality, there exists z Z[γ] such that z Ω x = [ x, c x ), z L and z / Ω y = [ y, c y ). In the case x < y, it yields x < c z y. In the case x > y, it yields y < z x. In either case, x and y 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.), Ω = [0, c). Output: The pallete of Σ(Ω). () Compute the set Ξ = {ξ 0 = 0 < ξ < < ξ N < ξ N = c} given by (5.6), with L := L c as defined in (5.5). (2) For each interval [ξ j, ξ j+ ) compute the local configuration of size L as Σ([ ξ j, c ξ j )) 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 c 0 > 0 and consider only values of c such that γ c 0 c < c 0. We are able to show that the pallete changes with c in a well-described way: Lemma 5.4. Let us fix b 0, c 0 R such that 0 < b 0 < c 0. Then there exists a finite set Θ = {θ < θ < < θ N } (b 0, c 0 ) such that the mapping c Pal ( [0, c) ) is constant on each of the intervals (θ j, θ j ) for j =,..., N, where we put θ 0 := b 0 and θ N := c 0. Proof. Consider L := L b0 defined in (5.5), and put (5.7) Θ := (Π 0 Π 0 ) (b 0, c 0 ), where Π 0 := { x Z[γ ] : x ( c 0, c 0 ) and x < L }. We also let Π := {x Z[γ ] : x B(0, L)}; clearly Π 0 = Π ( c 0, c 0 ). We will show that if for c, c 2 [b 0, c 0 ) there exists a protocell in the pallete for c that is not in the pallete for c 2, then necessarily a point of Θ lies between c and c 2.

10 0 T. HEJDA AND E. PELANTOVÁ x [ 0, ) x [, + γ ) x [ + γ, γ ) x [ γ, 2 + γ ) x [ 2 + γ, + γ ) x [ + γ, γ 2 ) x [ γ 2, + γ 2 ) Figure 2. Voronoi protocells for X 2 (γ) = Σ(Ω), where Ω = [0, 2/( γ )) and γ = γ T is the complex Tribonacci constant. Let us take such a protocell. It corresponds to a subset of Π of the form S := Π [ a, a + c ) with a [0, c ). Let us take the maximal interval (A, B) and the minimal interval [C, D] such that S = Π (A, B) = Π [C, D]. Then A, B, C, D Π, we have D C c B A and either c 2 D C or c 2 B A. In the first case c 2 D C, certainly C, D Π 0 therefore D C Θ. In the second case c 2 B A, we have 0 [ a, a + c ) (A, B), whence A < 0 and B > 0. Then B A c 2 gives A c 2 > c 0 and B c 2 < c 0, therefore A, B Π 0 and B A Θ. Let us apply the lemma in the case b 0 := γ c 0. It gives us all possible cut-points of the interval [γ c 0, c 0 ) 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.), c 0 > 0. Output: All possible palettes Pal(Ω) of Σ(Ω) for Ω = [0, c) and γ c 0 c < c 0. () Compute the set Θ = {θ 0 < θ < < θ N } given by (5.7), with L := L γ c 0 as defined in (5.5). (2) Using Algorithm 5.3, compute the palettes Pal(Ω) for all Ω = [0, c) with c = γ c 0, γ c 0+θ 0 2, θ 0, θ0+θ 2, θ,..., θ N 2+θ N 2, θ N, θ N +c 0 2. (3) Remove possible duplicates in the list of palettes. 6. Complex Tribonacci number exploited. Proof of Theorem.2 In this section, we will describe the details of the proposed workflow on an example the complex Tribonacci base γ = γ T. We aim at the proof of Theorem.2. We put := γγ = /γ in the sequel.

11 SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS Figure 3. Part of the Voronoi tessellation of X 2 (γ) = Σ(Ω), where Ω = [0, 2/( γ )) and γ = γ T 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.. Let Ω = [0, c) with c ( 2, 3 ), where γ is the complex Tribonacci constant and := /γ. Denote Σ := Σ(Ω). Then (6.) min δ( T (x) ) = / and max ( T (x) ) = 2 2 x Σ x Σ 3 2. Proof. We put c 0 := 3. Since Rγ < 0, we have that the argument of γ satisfies arg γ (π/2, π). Then arg γ 2 (π, 2π) and we have p = 2 for Lemma 5.. We can compute that k =. Then L = γ(γ ) Iγ (Since we use the symbolic computation in Sage, where elements of Q(γ) are stored in the form x = q 0 + q γ + q 2 γ 2 and q i are stored as ratios of integers, all the computations are carried out precisely; we evaluate the points a floating-point numbers only for the purpose of drawing the figures.) Using this L, we run Algorithm 5.5. This gives Θ of size 62. The number of cases in step 2 of this algorithm is then 23. This means that we have to run Algorithm 5.3 exactly 23 times to obtain all the possible palettes. One of the palletes, for c = 2 γ, i.e., for X 2 (γ), is depicted in Figure 2. In this case, the size of Ξ is 34, therefore we get 33 protocells. However, there are only 7 different ones after we remove all duplicates. We have drawn a part of the Voronoi tessellation of X 2 (γ) in Figure 3. Amongst the 23 cases mentioned above, there are many duplicates, and we end with only 8 cases. Moreover, we observe that for cut-points θ i, the palette is the

12 2 T. HEJDA AND E. PELANTOVÁ Interval for c The palette of Σ(Ω), where Ω = [0, c) 2 ( 2, 2) (2, + 2) ( + 2, 2 + ) ( 2 +, 2 + ) (2 +, 2 + ) ( 2 +, 2 + 2) ( 2 + 2, 2 + 2) (2 + 2, 3 ) Tile γ T 4 T γ T 5 T 2 T 3 γ T 8 T 4 T 5 T 6 T 7 T 8 γ T 0 T 9 T 0 Value of δ Value of A B A B B A B B B B B A B B Value of Table. The protocells for the complex Tribonacci constant for abritrary window Ω = [0, c). Each but the last tile in the list appears rotated by 80 as well, we omit these to make the table shorter. We omitt the palettes for the cut-points. However, a palette for a cut-point is the intersection of the palettes for the surrounding intervals, i.e., for instance Pal([0, 2 + )) = {T 2, T 6, T 8, T 9 }. We put A = and B = A. intersection of palletes of the two surrounding intervals. All the palette for the intervals are depicted in Table. At the bottom of the table, the values of δ(t ) and (T ) are written out for each protocell. It turns out that every row of the table but the special case c = 2 has minimal value of δ equal to / and maximal value of equal to This completes the proof of the proposition Proof of Theorem.2. The theorem is a direct corollary of the previous proposition and of the following two facts:

13 SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS 3 + Figure 4. Part of Voronoi (in solid lines) and Delone (in double lines) tessellations of X 2 (γ) for γ the complex Tribonacci constant. The white cross is a vertex of the Voronoi tessellation, and at the same time, it is a center of the gray circle, on which four points of X 2 (γ) lie. It cannot happen that c = m/( γ ) = (γγ) k = k for any m and k Z (which means we never hit the special case c = 2 in Table ). Let us suppose for contradiction that m/( γ ) = (γγ) k = /(γ ) k, i.e., m(γ ) k = γ. Since γ is cubic, we have necessarily k 3 of k 2. Case k 3: By the Galois isomorphism mγ k = γ. Then mγ k γ 3 > γ, contradiction. Case k 2: We have γ <, therefore m γ k γ 2 < γ, contradiction. We can use the self-similarity property: It is straightforward that δ(γ j T ) = γ j δ(t ) and similarly for ; this gives analogous result for the values of l and L. Proposition 6. corresponds to the case k = 2 of the theorem, since we have k = 2 if and only if c := m/( γ ) > (/γ ) 2 = 2 and c < (/γ ) 3 = 3. For this k, the values in (.4) and (6.) coincide (we remark that γ = and /γ = ). Remark. Let us point out that for a real base the characteristic L m () given by (.) is not influenced by gaps x k+ x k occurring only in a bounded piece of the real line. Therefore in general the value L m (γ) as we have defined for the complex number γ is not the precise analogy to L m (). 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 4 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 sup x 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 m (γ) is Delone, and L m(γ) = + 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 we get the following result: Theorem 7.. Let γ = γ T i be the complex root of the polynomial Y 3 + Y 2 + Y. Let m N. Find a maximal k Z such that m ( γ ) ( ) k, γ where γ is the real Galois conjugate of γ. Then we have L m(γ) = γ 3 k. 8. Comments and open problems This paper treats a family of cubic complex Pisot units γ such ones that the real number /γ 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: () We can consider a different perspective of the Tribonacci constant. Let γ be the complex root of Y 3 + Y 2 + Y, and put := /γ. Both γ and γ are complex Pisot units. It was shown by Vávra [Váv3] that the real Tribonacci constant has the so-called Property ( F). Shortly speaking, all numbers from I ), have a finite expansion of the form a + a2 2 + a3 3 + with a j {0, }. From this, we can show that X m ( γ) is a cut-and-project set for arbitrary m. The idea goes along the lines of the proof of Theorem 4.. (2) Consider any real Pisot unit of degree n. Let γ = i. Then γ is a complex Pisot unit of degree at most 2n, its Galois conjugates are γ and ±i for conjugates of. Z[ /] = I Z[], where I := ( +, +

15 SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS 5 Clearly X m (γ) = X m ( )+i X m ( ). Therefore the Voronoi cells of X m (γ) are rectangles. Values l m (γ) and L m (γ) can be easily obtained from the minimal and maximal distances in X m ( ). In the case n = 2, relations between X m ( ) and cut-and-project sets in dimensions d = e = were established in [MPP3], implying that X m (γ) is related to cut-and-project sets in dimensions d = e = 2. Let us note that Zaïmi [Zaï04] evaluated l m (γ) for γ = i, m = 2 and > the root of Y 2 ay a, a N. (3) In the cubic case, we can weaken the hypothesis of Theorem 4.. For a fixed m, the Property (F) can be replaced by the assumption that all numbers from Z[] [0, ) have a finite -representation over the alphabet {0,,..., m}, where we denote := /γ >. 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[]. They have shown that all of them are of the form.ccc =.c ω, where 0 c < and (a + b) c. Since all numbers from Z[γ] [0, ) have finite or periodic -expansions (and the only periods are therefore the ones mentioned above), it is satisfactory to find m such that the number.(a+b) ω has a finite representation over the alphabet {0,..., m }. Under this hypothesis, all numbers from Z[γ] [0, ) have a finite representation over the alphabet {0,..., m} for all m m a+b. 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 (, 2) are treated by Dombek, Masáková and Ziegler in [DMZ3]. 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 ±, then the question can be interpreted as Property (F) over the alphabet {, 0, }. 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 with a complex conjugate satisfy the following: There exists m N such that all number from Z[] [0, ) have finite -representation over the alphabet {0,..., m}? (B) Which real cubic bases, other than minus the Tribonacci constant, satisfy Property ( F)? (C) It is well known that in the real case, X m () is a relatively dense set in R + if and only if m >. Can we establish analogous result in the complex case? In particular, is X m (γ) relatively dense set in C for all m > γγ? Can the complex modification of the Feng s result [Fen3] be proved? Namely that l m (γ) = 0 if and only if m > γγ and γ is not a complex Pisot number? Acknowledgements We would like to thank Wolfgang Steiner for our fruitful discussions.

16 6 T. HEJDA AND E. PELANTOVÁ b a m Representation of.(a + b) ω 2 3 2a 2.(a 3)(2a 2)(a 3)(0)() 3 7 3a 6.(a 4)(2a 5)(3a 6)(a 7)(0)() = 6 0.(2)(7)(0)(0)(0)(0)() = 5 9.(0)(9)(9)(5)(0)(0)() = 4 7.(0)(2)(6)(7)(0) 3 () 4 8 8a.(a 5)(2a )(8a )(4a 3)(a 8)(0)() = 7 39.(0)(6)(39)(27)(0) 3 () = 6 47.(0)(3)(44)(47)(0) 4 () 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 and m m /γ a+b. This work was supported by Grant Agency of the Czech Technical University in Prague grant SGS/62/OHK4/3T/4, Czech Science Foundation grant S, and ANR/FWF project FAN Fractals and Numeration (ANR-2-IS0-0002, FWF grant I36). References [Aki00] Shigeki Akiyama, Cubic Pisot units with finite beta expansions, Algebraic number theory and Diophantine analysis (Graz, 998), de Gruyter, Berlin, 2000, pp. 26. MR (200i:095) [ARS04] Shigeki Akiyama, Hui Rao, and Wolfgang Steiner, A certain finiteness property of Pisot number systems, J. Number Theory 07 (2004), no., MR (2005g:35) [BH02] Peter Borwein and Kevin G. Hare, Some computations on the spectra of Pisot and Salem numbers, Math. Comp. 7 (2002), no. 238, MR (2003a:35) [BH03], General forms for minimal spectral values for a class of quadratic Pisot numbers, Bull. London Math. Soc. 35 (2003), no., MR (2003i:54) [Bug96] Yann Bugeaud, On a property of Pisot numbers and related questions, Acta Math. Hungar. 73 (996), no. -2, MR 4598 (98c:3) [DMZ3] Daniel Dombek, Zuzana Masáková, and Volker Ziegler, On distinct unit generated fields that are totally complex, 203, submitted, 4 pp. [EJK90] Paul Erdős, István Joó, and Vilmos Komornik, Characterization of the unique expansions = i= q n i and related problems, Bull. Soc. Math. France 8 (990), no. 3, MR (9j:006) [EJK98], On the sequence of numbers of the form ɛ 0 + ɛ q + + ɛ nq n, ɛ i {0, }, Acta Arith. 83 (998), no. 3, MR 685 (99a:022) [Fen3] De-Jun Feng, On the topology of polynomials with bounded integer coefficients, 203, preprint. [FW02] De-Jun Feng and Zhi-Ying Wen, A property of Pisot numbers, J. Number Theory 97 (2002), no. 2, MR (2003i:55) [KLP00] Vilmos Komornik, Paola Loreti, and Marco Pedicini, An approximation property of Pisot numbers, J. Number Theory 80 (2000), no. 2, MR (2000k:6) [Kom02] Takao Komatsu, An approximation property of quadratic irrationals, Bull. Soc. Math. France 30 (2002), no., MR 9069 (2003b:063)

17 SPECTRAL PROPERTIES OF CUBIC COMPLEX PISOT UNITS 7 [Moo97] Robert V. Moody, Meyer sets and their duals, The mathematics of long-range aperiodic order (Waterloo, ON, 995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 489, Kluwer Acad. Publ., Dordrecht, 997, pp MR (98e:52029) [MPP3] Zuzana Masáková, Kateřina Pastirčáková, and Edita Pelantová, 203, 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, MR (2004a:5204) [MPZ03b], Classification of Voronoi and Delone tiles of quasicrystals. II. Circular acceptance window of arbitrary size, J. Phys. A 36 (2003), no. 7, MR (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, MR (2005j:52025) [Sage] The Sage Group, Sage: Open source mathematical software (version 5.0), 203, http: // [TikZ] Till Tantau et al., TikZ & PGF (version 2.0), 200, projects/pgf. [Váv3] Tomáš Vávra, Arithmetical aspects of a number system with negative Tribonacci base, Doktorandské dny (Praha, 203), Czech Technical University in Prague, 203, http: //kmwww.fjfi.cvut.cz/ddny/?loc=historie, pp [Zaï04] Toufik Zaïmi, On an approximation property of Pisot numbers. II, J. Théor. Nombres Bordeaux 6 (2004), no., MR (2006f:33) Dept. of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 3, Prague 2000, Czech Rep. Current address: LIAFA, CNRS UMR 7089, Université Paris Diderot Paris 7, Case 704, Paris Cedex 3, France address: tohecz@gmail.com Dept. of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 3, Prague 2000, Czech Rep. address: edita.pelantova@fjfi.cvut.cz