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Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II
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Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II Alan Haynes a , ∗, Michael Kelly b , Henna Koivusalo c a University of Houston, Houston, TX 77204, USA b Center for Communications Research, Princeton, NJ 08540, USA c Faculty of Mathematics, University of Vienna, Oskar Morgensternplatz 1, 1090 Vienna, Austria
Abstract Recent results of several authors have led to constructions of parallelotopes which are bounded remainder sets for totally irrational toral rotations. In this brief note we explain, in retrospect, how some of these results can easily be obtained from a geometric argument which was previously employed by Duneau and Oguey in the study of deformation properties of mathematical models for quasicrystals. c 2016 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝ Keywords: Bounded remainder sets; Cut and project sets; Quasicrystals
1. Introduction Let T : Ts → Ts be a Lebesgue measure preserving transformation of the s-dimensional torus Ts = Rs /Zs . A measurable set A ⊆ Rs is a bounded remainder set (henceforth denoted BRS) for T if ⏐ N −1 ⏐ ⏐∑ ⏐ ⏐ ⏐ n sup sup ⏐ χ A (T (x)) − N |A|⏐ < ∞, ⏐ x∈Rs N ∈N ⏐ n=0
where χ A : T → Z≥0 is the indicator function of A, viewed as a multi-set in Ts (i.e. so that χ A could potentially take any non-negative integer value), and |A| denotes its measure. Throughout ∗ Correspondence to: Department of Mathematics University of Houston, Houston, TX 77204-3008, United States.
E-mail addresses: [email protected] (A. Haynes), [email protected] (M. Kelly), [email protected] (H. Koivusalo). http://dx.doi.org/10.1016/j.indag.2016.11.010 c 2016 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝
Please cite this article in press as: A. Haynes, et al., Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II, Indagationes Mathematicae (2016), http://dx.doi.org/10.1016/j.indag.2016.11.010
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this paper we will assume that T is a toral rotation, given by T (x) = x + α for some α ∈ Rs . This is a situation which is particularly important in Diophantine approximation, and over the course of the past century it has been studied by a number of authors. For s = 1 the problem of classifying BRS’s is satisfactorily dealt with by works of Hecke [11], Ostrowski [17,18], and Kesten [13] (see also related results of Oren [16]), which together show that for an irrational rotation of T by α, a necessary and sufficient condition for an interval I to be a BRS is that |I| ∈ αZ + Z. Several papers [15,22–25] have investigated the corresponding problems in higher dimensions. Of particular note are the works of Sz¨usz [22], who demonstrated a construction of parallelogram BRS’s when s = 2, and Liardet [15, Theorem 4], who used a dynamical cocycles argument to extend Sz¨usz’s construction to arbitrary s > 1. Other significant connections between BRS’s and dynamical systems have been highlighted in [8,10,12,19], and a more thorough exposition of the history of these sets can be found in the introduction of [9]. Our understanding of polytope BRS’s in higher dimensions has recently been substantially improved by Grepstad and Lev in [9]. One of their central results is the following theorem. Theorem 1.1. Suppose that α ∈ Rs and that 1, α1 , . . ., and αs are linearly independent over Q. Then for any choice of linearly independent vectors v1 , . . . , vs ∈ αZ + Zs , the parallelotope ⎧ ⎫ s ⎨∑ ⎬ P= tjvj : 0 ≤ tj < 1 (1.1) ⎭ ⎩ j=1
is a BRS for the rotation of Ts by α. Grepstad and Lev’s proof of Theorem 1.1 involved a detailed Fourier analytic argument which allowed them to calculate the ‘transfer functions’ of the parallelotopes P. One of the goals of this paper is to explain how the theorem can be deduced using a simple and elegant geometric argument, first introduced by Duneau and Oguey [6] in the study of regularity properties of quasicrystalline materials. When applied to the BRS problem, the essence of their argument shows that the collection of return times of a toral rotation to a region of the form (1.1) has a group structure coming from a natural higher dimensional realization of the problem. Our main theorem, which is an abstract but otherwise straightforward generalization of [6, Theorem 3.1], is Theorem 2.1. Upon first reading, the statement of the theorem may seem rather complicated. Our reason for presenting it in such a form is to highlight its flexibility in terms of potential applications. The theorem applies not only to orbits of toral rotations, which can be recast as one dimensional quasicrystals, but also to multidimensional point patterns in Euclidean space. As a particular example, the following result will be shown to be an easy consequence of our main result. Theorem 1.2. Let Y ⊆ R2 be the collection of vertices of any Penrose tiling. Then there is a bijection from Y to a lattice, which moves every point by at most a uniformly finite amount. This result was mentioned in the paper of Duneau and Oguey (see comments on [6, p.13]), and it has recently been proved using different means by Solomon [21]. It was pointed out to us by the referee that an earlier proof idea can be found in [5], and later developments appear in [1]. We wish to emphasize that Theorem 1.2 is only representative of the potential applications of our main theorem. The same argument used in its proof applies to any generic canonical cut and project set in Euclidean space. However, to avoid a long list of definitions and technical points regarding these objects, we maintain focus on Penrose tilings, viewed as linear transformations
Please cite this article in press as: A. Haynes, et al., Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II, Indagationes Mathematicae (2016), http://dx.doi.org/10.1016/j.indag.2016.11.010
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of 5 to 2 cut and project sets. For readers interested in learning about more general cut and project sets, we refer to the excellent book [2] by Baake and Grimm. It is also worth noting that the hypotheses Theorem 2.1 allow us the flexibility to formulate results in many interesting non-Euclidean spaces. This is a topic which we believe will lead to new results, and which we leave open for further exploration. 2. Statement and proof of main theorem Let X be a metrizable Q-vector space with a translation invariant metric d : X × X → [0, ∞), and suppose that V p and Vi are complementary Q-vector subspaces of X , in the sense that V p ∩ Vi = {0} and X = V p + Vi , where the right hand side denotes the Minkowski sum of the two sets. In all of what follows, subsets of X will be taken with the usual induced topology. Let ρ p and ρi be the projections according to the above decomposition, onto V p and Vi , respectively. Suppose that Γ is a subgroup of X which is also a finite dimensional Z-module. Given a relatively compact W ⊆ Vi , we define a set Y˜ ⊆ X and a multi-set Y ⊆ V p by Y˜ = ρi−1 (W ) ∩ Γ
and
Y = ρ p (Y˜ ).
We emphasize that Y is a multi-set, so that if the restriction of ρ p to Y˜ is not injective, the elements of Y are listed with the appropriate multiplicity. Theorem 2.1. With the notation above, suppose further that: (i) Z is a subspace of X (possibly different than Vi ) which is complementary to V p , φ p : X → V p is the projection onto V p with respect to the decomposition X = V p + Z , (ii) The group Γ has a decomposition of the form Γ = Λ + Λc , where Λ and Λc are groups with the properties that Λ ∩ Λc = {0} and Z ∩ Γ = Λ, and (iii) W is the image under ρi of a relatively compact fundamental domain for Z /Λ′ , where Λ′ ⩽ Λ and [Λ : Λ′ ] = N < ∞. Then there is a bijection f from Y to a subgroup Λ′c of V p satisfying φ p (Λc ) ⩽ Λ′c and : φ p (Λc )] = N , with the property that f moves every point of Y by at most a uniformly finite amount. [Λ′c
Proof. From assumption (ii) it follows that Y˜ can be written as the disjoint union ⨆ Y˜ = (Z + λ) ∩ Y˜ . λ∈Λc
Let W Z = ρi−1 (W ) ∩ Z . For λ ∈ Λc consider the set Wλ = (Z + λ) ∩ ρi−1 (W ) − λ. Each such set is a translate of W Z in Z , and therefore a fundamental domain for Z /Λ′ . It follows that each set Wλ contains exactly N points of Λ, and from this we have that |(Z + λ) ∩ Y˜ | = N .
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For each λ ∈ Λc let us write (Z + λ) ∩ Y˜ = {yλ (0), . . . , yλ (N − 1)}. Choose a Z-basis γ1 , . . . , γd for Λc , and for 1 ≤ j ≤ d let γ j′ = φ p (γ j ). From the assumptions in hypothesis (ii), it is easy to deduce that the restriction of the map φ p to Λc is injective. It follows that the elements N1 γ1′ , γ2′ , . . . , γd′ generate a group Λ′c ⩽ V p which contains φ p (Λc ) as a subgroup of index N . Note that the Q-vector space structure is what allows us to divide by N . Now, using the decomposition Y˜ =
−1 ⨆ N⨆
yλ ( j),
λ∈Λc j=0
we define a bijection g : Y˜ → Λ′c by j ′ γ, N 1 and we use this to define f : Y → Λ′c by g(yλ ( j)) = φ p (λ) +
˜ f (y) = g(ρ −1 p (y) ∩ Y ). Recall that Y is a multi-set, so the map f does produce a well defined bijection between Y and Λ′c . All that remains is to show that f moves every point by at most a uniformly finite amount. ˜ Let y ∈ Y and write y˜ = ρ −1 p (y) ∩ Y in two different ways, as y˜ = vi + v (1) p ,
where vi ∈ Vi and v (1) p ∈ Vp ,
and y˜ = z + v (2) p ,
where z ∈ Z and v (2) p ∈ Vp ,
′ (2) so that y = v (1) p and f (y) = v p + ( j/N )γ1 , for some 0 ≤ j < N . Notice that, because of the definition of Y˜ , we have that
vi ∈ W
and
z ∈ WZ .
Therefore we have that ( ) j d( f (y), y) ≤ d y˜ − z + γ1′ , y˜ − z + d( y˜ − vi , y˜ − z) N ( ) j ′ ≤d γ , 0 + sup sup d(−v, −z ′ ). N 1 v∈W z ′ ∈W Z Since W ∪ W Z is a relatively compact subset of X , the right hand side of this inequality is finite. Since it does not depend on y, the conclusion follows. □ 3. Proof of Theorem 1.1 For the proof of Theorem 1.1 we will apply Theorem 2.1 with X = Rs+1 , Γ = Zs+1 , V p = ⟨(α, 1)⟩R ,
and
Vi = ⟨e1 , . . . , es ⟩R ,
Please cite this article in press as: A. Haynes, et al., Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II, Indagationes Mathematicae (2016), http://dx.doi.org/10.1016/j.indag.2016.11.010
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where e j denotes the jth standard basis vector for Rs+1 . Write each of the vectors v j from the statement of Theorem 1.1 as ( j)
v j = n ( j) − n s+1 α, ( j)
with n s+1 ∈ Z and n ( j) ∈ Zs , and let W = P − x where x ∈ Vi and P is the parallelotope of ( j) (1.1), realized as a subset of Vi . If we set λ j = (n ( j) , n s+1 ) ∈ Γ then for each j we have that v j = ρi (λ j ). We take Z to be the real subspace of X generated by the vectors λ1 , . . . , λs , and we set Λ = Z ∩Γ . It is a straightforward exercise to show that we can find a vector λc ̸∈ Z for which Γ = Λ + Λc ,
with Λc = ⟨λc ⟩Z .
Therefore the hypotheses of Theorem 2.1 are satisfied, allowing us to conclude that the corresponding multi-set Y is in bijection, via a bounded distance map, with a lattice in V p . Now let Y ′ be the multi-set defined by Y ′ = {n ∈ Z : nα ∈ P − x mod Zs } ⊆ R, where each integer n ∈ Y ′ is counted with multiplicity equal to the number of integer vectors m ∈ Zs for which nα + m ∈ P − x. Applying a linear transformation to X in our above argument, we deduce that Y ′ is in bounded distance bijection with a set of the form γ Z ⊆ R, for some γ > 0. Since 1, α1 , . . ., and αs are Q-linearly independent, the Birkhoff Ergodic Theorem (alternatively, Weyl’s criterion from uniform distribution theory) implies that γ = |P|−1 . Write Y ′ = {n j } j∈Z , with n 0 ≤ 0 < n 1 and n j ≤ n j+1 for all j. Given N ∈ N, choose K ≥ 0 so that n K ≤ N < n K +1 . Then we have N ∑ χ P (nα + x) = K , n=1
and also that K = |P|n K + O(1) = |P|N + O(1). After rearranging terms and taking the supremum over all x, this shows that P is a bounded remainder set. 4. Proof of Theorem 1.2 The argument we will use in our proof of Theorem 1.2 closely follows that outlined in [6, p.13], the only difference being that we do not appeal to [6, Theorem 4.1], but instead use the following result due to Laczkovich (see [14, Theorem 1.1]). Theorem 4.1. For any discrete set S ⊆ Rd and for any κ > 0 the following conditions are equivalent: (i) There is a positive constant C > 0 with the property that, for any set C ⊆ Rd which is a finite union of unit cubes, we have that |#(S ∩ C) − κ|C|d | ≤ C · |∂C|d−1 , where | · |k denotes k-dimensional Lebesgue measure.
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(ii) There is a bijection from S to α −1/d Zd , which moves every point by at most a uniformly finite amount. Let ζ = exp(2πi/5), take X = R5 , and let V p be the two dimensional real subspace of X generated by the vectors (1, Re(ζ ), Re(ζ 2 ), Re(ζ 3 ), Re(ζ 4 )) and (0, Im(ζ ), Im(ζ 2 ), Im(ζ 3 ), Im(ζ 4 )). Take Vi to be the real subspace of X orthogonal to V p , and let W ⊆ Vi be a translate of the image under ρi of the unit cube in R5 . Well known results of de Bruijn [4] and Robinson [20], further developed in [3] and [7], show that the set Y obtained in this way is the image under a linear transformation of the collection of vertices of a Penrose tiling, and in fact that all Penrose tilings can be obtained in a similar way. () The set W can be written as a disjoint union of 53 = 10 parallelotopes W1 , . . . , W10 , each of which is a translate of the projection to Vi of a parallelotope in R5 generated by 3 elementary basis vectors. Each parallelotope W j satisfies the hypotheses of Theorem 2.1, and therefore gives rise to a set Y j which is in bijection, via a bounded distance map, with a lattice. Appealing to Theorem 4.1, for each j and for any region C ∈ V p which is a finite union of unit cubes (chosen with respect to some basis for V p ), the number of points in Y j ∩ C is equal to the expected number plus an error which is at most a constant multiple of the volume of the boundary. Therefore, by summing over all j, we see that the set Y also satisfies condition (i) of Theorem 4.1, and the conclusion of Theorem 1.2 follows immediately. Acknowledgments AH wishes to thank Dirk Frettl¨oh for helpful conversations in Delft regarding the context of the results in this paper, and for sharing a preprint (joint work between him and Alexey Garber) in which similar results have been obtained. The authors also thank the referee for carefully reading the manuscript and providing useful feedback. The first author’s research was supported by EPSRC grants L001462, J00149X, M023540. The third author’s research was supported by Osk. Huttunen Foundation. The second author’s research was supported by NSF grants DMS-1101326, DMS-1045119 and DMS-0943832. References [1] J. Aliste-Prieto, D. Coronel, J.M. Gambaudo, Linearly repetitive Delone sets are rectifiable, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2) (2013) 275–290. [2] M. Baake, U. Grimm, Aperiodic order. Vol. 1. A mathematical invitation, Encyclopedia of Mathematics and its Applications, vol. 149, Cambridge, 2013. [3] M. Baake, P. Kramer, M. Schlottmann, D. Zeidler, Planar patterns with fivefold symmetry as sections of periodic structures in 4-space, Internat. J. Modern Phys. B 4 (15–16) (1990) 2217–2268. [4] N.G. de Bruijn, Algebraic theory of Penrose’s nonperiodic tilings of the plane, I, II, Indag. Math. (N.S.) 43 (1) (1981) 39–52, 53-66. [5] W.A. Deuber, M. Simonovits, V.T. Sós, A note on paradoxical metric spaces, Studia Sci. Math. Hungar. 30 (1–2) (1995) 17–23. [6] M. Duneau, C. Oguey, Displacive transformations and quasicrystalline symmetries, J. Physique 51 (1) (1990) 5–19. [7] F. Gähler, J. Rhyner, Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings, J. Phys. A 19 (2) (1986) 267–277.
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[8] P.J. Grabner, P. Hellekalek, P. Liardet, The dynamical point of view of low-discrepancy sequences, Unif. Distrib. Theory 7 (1) (2012) 11–70. [9] S. Grepstad, N. Lev, Sets of bounded discrepancy for multi-dimensional irrational rotation, Geom. Funct. Anal. 25 (1) (2015) 87–133. [10] A. Haynes, H. Koivusalo, Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, Israel J. Math. 212 (1) (2016) 189–201. [11] E. Hecke, Über analytische Funktionen und die Verteilung von Zahlen mod. eins., (German), Abh. Math. Semin. Univ. Hambg 1 (1) (1922) 54–76. [12] M. Kelly, L. Sadun, Pattern equivariant cohomology and theorems of Kesten and Oren, Bull. Lond. Math. Soc. 47 (1) (2015) 13–20. [13] H. Kesten, On a conjecture of Erd˝os and Szüsz related to uniform distribution mod 1, Acta Arith. 12 (1966/1967) 193–212. [14] M. Laczkovich, Uniformly spread discrete sets in R d , J. Lond. Math. Soc. (2) 46 (1) (1992) 39–57. [15] P. Liardet, Regularities of distribution, Compos. Math. 61 (3) (1987) 267–293. [16] I. Oren, Admissible functions with multiple discontinuities, Israel J. Math. 42 (4) (1982) 353–360. [17] A. Ostrowski, Math. Miszellen IX, Notiz zur Theorie der Diophantischen Approximationen, Jahresber. Dtsch. Math.-Ver. 36 (1927) 178–180. [18] A. Ostrowski, Math. Miszellen XVI, Zur Theorie der linearen Diophantischen Approximationen, Jahresber. Dtsch. Math.-Ver. 39 (1930) 34–46. [19] G. Rauzy, Ensembles à restes bornés, in: Seminaire De Théorie Des Nombres De Bordeaux, 1972, pp. 1–12. [20] E.A. Robinson Jr., The dynamical properties of Penrose tilings, Trans. Amer. Math. Soc. 348 (11) (1996) 4447–4464. [21] Y. Solomon, Substitution tilings and separated nets with similarities to the integer lattice, Israel J. Math. 181 (2011) 445–460. [22] P. Szüsz, Über die Verteilung der Vielfachen einer komplexen Zahl nach dem Modul des Einheitsquadrats, (German), Acta Math. Acad. Sci. Hungar. 5 (1954) 35–39. [23] V.G. Zhuravlev, Rauzy tilings and bounded remainder sets on a torus, in: Trudy po Teorii Chisel, 83-106, 253, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 322 (2005) (Russian); translation in J. Math. Sci. (N. Y.) 137 (2) (2006) 4658–4672. [24] V.G. Zhuravlev, Exchanged toric developments and bounded remainder sets, in: Analiticheskaya Teoriya Chisel i Teoriya Funktsii. 26, 95–145, 219–220, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 392 (2011) (Russian); translation in J. Math. Sci. (N.Y.) 184 (6) (2012) 716–745. [25] V.G. Zhuravlev, Bounded remainder polyhedra, Proc. Steklov Inst. Math. 280 (suppl. 2) (2013) S71–S90.
Please cite this article in press as: A. Haynes, et al., Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II, Indagationes Mathematicae (2016), http://dx.doi.org/10.1016/j.indag.2016.11.010
ScienceDirect Indagationes Mathematicae (
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Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II Alan Haynes a , ∗, Michael Kelly b , Henna Koivusalo c a University of Houston, Houston, TX 77204, USA b Center for Communications Research, Princeton, NJ 08540, USA c Faculty of Mathematics, University of Vienna, Oskar Morgensternplatz 1, 1090 Vienna, Austria
Abstract Recent results of several authors have led to constructions of parallelotopes which are bounded remainder sets for totally irrational toral rotations. In this brief note we explain, in retrospect, how some of these results can easily be obtained from a geometric argument which was previously employed by Duneau and Oguey in the study of deformation properties of mathematical models for quasicrystals. c 2016 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝ Keywords: Bounded remainder sets; Cut and project sets; Quasicrystals
1. Introduction Let T : Ts → Ts be a Lebesgue measure preserving transformation of the s-dimensional torus Ts = Rs /Zs . A measurable set A ⊆ Rs is a bounded remainder set (henceforth denoted BRS) for T if ⏐ N −1 ⏐ ⏐∑ ⏐ ⏐ ⏐ n sup sup ⏐ χ A (T (x)) − N |A|⏐ < ∞, ⏐ x∈Rs N ∈N ⏐ n=0
where χ A : T → Z≥0 is the indicator function of A, viewed as a multi-set in Ts (i.e. so that χ A could potentially take any non-negative integer value), and |A| denotes its measure. Throughout ∗ Correspondence to: Department of Mathematics University of Houston, Houston, TX 77204-3008, United States.
E-mail addresses: [email protected] (A. Haynes), [email protected] (M. Kelly), [email protected] (H. Koivusalo). http://dx.doi.org/10.1016/j.indag.2016.11.010 c 2016 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝
Please cite this article in press as: A. Haynes, et al., Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II, Indagationes Mathematicae (2016), http://dx.doi.org/10.1016/j.indag.2016.11.010
2
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)
–
this paper we will assume that T is a toral rotation, given by T (x) = x + α for some α ∈ Rs . This is a situation which is particularly important in Diophantine approximation, and over the course of the past century it has been studied by a number of authors. For s = 1 the problem of classifying BRS’s is satisfactorily dealt with by works of Hecke [11], Ostrowski [17,18], and Kesten [13] (see also related results of Oren [16]), which together show that for an irrational rotation of T by α, a necessary and sufficient condition for an interval I to be a BRS is that |I| ∈ αZ + Z. Several papers [15,22–25] have investigated the corresponding problems in higher dimensions. Of particular note are the works of Sz¨usz [22], who demonstrated a construction of parallelogram BRS’s when s = 2, and Liardet [15, Theorem 4], who used a dynamical cocycles argument to extend Sz¨usz’s construction to arbitrary s > 1. Other significant connections between BRS’s and dynamical systems have been highlighted in [8,10,12,19], and a more thorough exposition of the history of these sets can be found in the introduction of [9]. Our understanding of polytope BRS’s in higher dimensions has recently been substantially improved by Grepstad and Lev in [9]. One of their central results is the following theorem. Theorem 1.1. Suppose that α ∈ Rs and that 1, α1 , . . ., and αs are linearly independent over Q. Then for any choice of linearly independent vectors v1 , . . . , vs ∈ αZ + Zs , the parallelotope ⎧ ⎫ s ⎨∑ ⎬ P= tjvj : 0 ≤ tj < 1 (1.1) ⎭ ⎩ j=1
is a BRS for the rotation of Ts by α. Grepstad and Lev’s proof of Theorem 1.1 involved a detailed Fourier analytic argument which allowed them to calculate the ‘transfer functions’ of the parallelotopes P. One of the goals of this paper is to explain how the theorem can be deduced using a simple and elegant geometric argument, first introduced by Duneau and Oguey [6] in the study of regularity properties of quasicrystalline materials. When applied to the BRS problem, the essence of their argument shows that the collection of return times of a toral rotation to a region of the form (1.1) has a group structure coming from a natural higher dimensional realization of the problem. Our main theorem, which is an abstract but otherwise straightforward generalization of [6, Theorem 3.1], is Theorem 2.1. Upon first reading, the statement of the theorem may seem rather complicated. Our reason for presenting it in such a form is to highlight its flexibility in terms of potential applications. The theorem applies not only to orbits of toral rotations, which can be recast as one dimensional quasicrystals, but also to multidimensional point patterns in Euclidean space. As a particular example, the following result will be shown to be an easy consequence of our main result. Theorem 1.2. Let Y ⊆ R2 be the collection of vertices of any Penrose tiling. Then there is a bijection from Y to a lattice, which moves every point by at most a uniformly finite amount. This result was mentioned in the paper of Duneau and Oguey (see comments on [6, p.13]), and it has recently been proved using different means by Solomon [21]. It was pointed out to us by the referee that an earlier proof idea can be found in [5], and later developments appear in [1]. We wish to emphasize that Theorem 1.2 is only representative of the potential applications of our main theorem. The same argument used in its proof applies to any generic canonical cut and project set in Euclidean space. However, to avoid a long list of definitions and technical points regarding these objects, we maintain focus on Penrose tilings, viewed as linear transformations
Please cite this article in press as: A. Haynes, et al., Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II, Indagationes Mathematicae (2016), http://dx.doi.org/10.1016/j.indag.2016.11.010
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A. Haynes et al. / Indagationes Mathematicae (
)
–
of 5 to 2 cut and project sets. For readers interested in learning about more general cut and project sets, we refer to the excellent book [2] by Baake and Grimm. It is also worth noting that the hypotheses Theorem 2.1 allow us the flexibility to formulate results in many interesting non-Euclidean spaces. This is a topic which we believe will lead to new results, and which we leave open for further exploration. 2. Statement and proof of main theorem Let X be a metrizable Q-vector space with a translation invariant metric d : X × X → [0, ∞), and suppose that V p and Vi are complementary Q-vector subspaces of X , in the sense that V p ∩ Vi = {0} and X = V p + Vi , where the right hand side denotes the Minkowski sum of the two sets. In all of what follows, subsets of X will be taken with the usual induced topology. Let ρ p and ρi be the projections according to the above decomposition, onto V p and Vi , respectively. Suppose that Γ is a subgroup of X which is also a finite dimensional Z-module. Given a relatively compact W ⊆ Vi , we define a set Y˜ ⊆ X and a multi-set Y ⊆ V p by Y˜ = ρi−1 (W ) ∩ Γ
and
Y = ρ p (Y˜ ).
We emphasize that Y is a multi-set, so that if the restriction of ρ p to Y˜ is not injective, the elements of Y are listed with the appropriate multiplicity. Theorem 2.1. With the notation above, suppose further that: (i) Z is a subspace of X (possibly different than Vi ) which is complementary to V p , φ p : X → V p is the projection onto V p with respect to the decomposition X = V p + Z , (ii) The group Γ has a decomposition of the form Γ = Λ + Λc , where Λ and Λc are groups with the properties that Λ ∩ Λc = {0} and Z ∩ Γ = Λ, and (iii) W is the image under ρi of a relatively compact fundamental domain for Z /Λ′ , where Λ′ ⩽ Λ and [Λ : Λ′ ] = N < ∞. Then there is a bijection f from Y to a subgroup Λ′c of V p satisfying φ p (Λc ) ⩽ Λ′c and : φ p (Λc )] = N , with the property that f moves every point of Y by at most a uniformly finite amount. [Λ′c
Proof. From assumption (ii) it follows that Y˜ can be written as the disjoint union ⨆ Y˜ = (Z + λ) ∩ Y˜ . λ∈Λc
Let W Z = ρi−1 (W ) ∩ Z . For λ ∈ Λc consider the set Wλ = (Z + λ) ∩ ρi−1 (W ) − λ. Each such set is a translate of W Z in Z , and therefore a fundamental domain for Z /Λ′ . It follows that each set Wλ contains exactly N points of Λ, and from this we have that |(Z + λ) ∩ Y˜ | = N .
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For each λ ∈ Λc let us write (Z + λ) ∩ Y˜ = {yλ (0), . . . , yλ (N − 1)}. Choose a Z-basis γ1 , . . . , γd for Λc , and for 1 ≤ j ≤ d let γ j′ = φ p (γ j ). From the assumptions in hypothesis (ii), it is easy to deduce that the restriction of the map φ p to Λc is injective. It follows that the elements N1 γ1′ , γ2′ , . . . , γd′ generate a group Λ′c ⩽ V p which contains φ p (Λc ) as a subgroup of index N . Note that the Q-vector space structure is what allows us to divide by N . Now, using the decomposition Y˜ =
−1 ⨆ N⨆
yλ ( j),
λ∈Λc j=0
we define a bijection g : Y˜ → Λ′c by j ′ γ, N 1 and we use this to define f : Y → Λ′c by g(yλ ( j)) = φ p (λ) +
˜ f (y) = g(ρ −1 p (y) ∩ Y ). Recall that Y is a multi-set, so the map f does produce a well defined bijection between Y and Λ′c . All that remains is to show that f moves every point by at most a uniformly finite amount. ˜ Let y ∈ Y and write y˜ = ρ −1 p (y) ∩ Y in two different ways, as y˜ = vi + v (1) p ,
where vi ∈ Vi and v (1) p ∈ Vp ,
and y˜ = z + v (2) p ,
where z ∈ Z and v (2) p ∈ Vp ,
′ (2) so that y = v (1) p and f (y) = v p + ( j/N )γ1 , for some 0 ≤ j < N . Notice that, because of the definition of Y˜ , we have that
vi ∈ W
and
z ∈ WZ .
Therefore we have that ( ) j d( f (y), y) ≤ d y˜ − z + γ1′ , y˜ − z + d( y˜ − vi , y˜ − z) N ( ) j ′ ≤d γ , 0 + sup sup d(−v, −z ′ ). N 1 v∈W z ′ ∈W Z Since W ∪ W Z is a relatively compact subset of X , the right hand side of this inequality is finite. Since it does not depend on y, the conclusion follows. □ 3. Proof of Theorem 1.1 For the proof of Theorem 1.1 we will apply Theorem 2.1 with X = Rs+1 , Γ = Zs+1 , V p = ⟨(α, 1)⟩R ,
and
Vi = ⟨e1 , . . . , es ⟩R ,
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where e j denotes the jth standard basis vector for Rs+1 . Write each of the vectors v j from the statement of Theorem 1.1 as ( j)
v j = n ( j) − n s+1 α, ( j)
with n s+1 ∈ Z and n ( j) ∈ Zs , and let W = P − x where x ∈ Vi and P is the parallelotope of ( j) (1.1), realized as a subset of Vi . If we set λ j = (n ( j) , n s+1 ) ∈ Γ then for each j we have that v j = ρi (λ j ). We take Z to be the real subspace of X generated by the vectors λ1 , . . . , λs , and we set Λ = Z ∩Γ . It is a straightforward exercise to show that we can find a vector λc ̸∈ Z for which Γ = Λ + Λc ,
with Λc = ⟨λc ⟩Z .
Therefore the hypotheses of Theorem 2.1 are satisfied, allowing us to conclude that the corresponding multi-set Y is in bijection, via a bounded distance map, with a lattice in V p . Now let Y ′ be the multi-set defined by Y ′ = {n ∈ Z : nα ∈ P − x mod Zs } ⊆ R, where each integer n ∈ Y ′ is counted with multiplicity equal to the number of integer vectors m ∈ Zs for which nα + m ∈ P − x. Applying a linear transformation to X in our above argument, we deduce that Y ′ is in bounded distance bijection with a set of the form γ Z ⊆ R, for some γ > 0. Since 1, α1 , . . ., and αs are Q-linearly independent, the Birkhoff Ergodic Theorem (alternatively, Weyl’s criterion from uniform distribution theory) implies that γ = |P|−1 . Write Y ′ = {n j } j∈Z , with n 0 ≤ 0 < n 1 and n j ≤ n j+1 for all j. Given N ∈ N, choose K ≥ 0 so that n K ≤ N < n K +1 . Then we have N ∑ χ P (nα + x) = K , n=1
and also that K = |P|n K + O(1) = |P|N + O(1). After rearranging terms and taking the supremum over all x, this shows that P is a bounded remainder set. 4. Proof of Theorem 1.2 The argument we will use in our proof of Theorem 1.2 closely follows that outlined in [6, p.13], the only difference being that we do not appeal to [6, Theorem 4.1], but instead use the following result due to Laczkovich (see [14, Theorem 1.1]). Theorem 4.1. For any discrete set S ⊆ Rd and for any κ > 0 the following conditions are equivalent: (i) There is a positive constant C > 0 with the property that, for any set C ⊆ Rd which is a finite union of unit cubes, we have that |#(S ∩ C) − κ|C|d | ≤ C · |∂C|d−1 , where | · |k denotes k-dimensional Lebesgue measure.
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(ii) There is a bijection from S to α −1/d Zd , which moves every point by at most a uniformly finite amount. Let ζ = exp(2πi/5), take X = R5 , and let V p be the two dimensional real subspace of X generated by the vectors (1, Re(ζ ), Re(ζ 2 ), Re(ζ 3 ), Re(ζ 4 )) and (0, Im(ζ ), Im(ζ 2 ), Im(ζ 3 ), Im(ζ 4 )). Take Vi to be the real subspace of X orthogonal to V p , and let W ⊆ Vi be a translate of the image under ρi of the unit cube in R5 . Well known results of de Bruijn [4] and Robinson [20], further developed in [3] and [7], show that the set Y obtained in this way is the image under a linear transformation of the collection of vertices of a Penrose tiling, and in fact that all Penrose tilings can be obtained in a similar way. () The set W can be written as a disjoint union of 53 = 10 parallelotopes W1 , . . . , W10 , each of which is a translate of the projection to Vi of a parallelotope in R5 generated by 3 elementary basis vectors. Each parallelotope W j satisfies the hypotheses of Theorem 2.1, and therefore gives rise to a set Y j which is in bijection, via a bounded distance map, with a lattice. Appealing to Theorem 4.1, for each j and for any region C ∈ V p which is a finite union of unit cubes (chosen with respect to some basis for V p ), the number of points in Y j ∩ C is equal to the expected number plus an error which is at most a constant multiple of the volume of the boundary. Therefore, by summing over all j, we see that the set Y also satisfies condition (i) of Theorem 4.1, and the conclusion of Theorem 1.2 follows immediately. Acknowledgments AH wishes to thank Dirk Frettl¨oh for helpful conversations in Delft regarding the context of the results in this paper, and for sharing a preprint (joint work between him and Alexey Garber) in which similar results have been obtained. The authors also thank the referee for carefully reading the manuscript and providing useful feedback. The first author’s research was supported by EPSRC grants L001462, J00149X, M023540. The third author’s research was supported by Osk. Huttunen Foundation. The second author’s research was supported by NSF grants DMS-1101326, DMS-1045119 and DMS-0943832. References [1] J. Aliste-Prieto, D. Coronel, J.M. Gambaudo, Linearly repetitive Delone sets are rectifiable, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2) (2013) 275–290. [2] M. Baake, U. Grimm, Aperiodic order. Vol. 1. A mathematical invitation, Encyclopedia of Mathematics and its Applications, vol. 149, Cambridge, 2013. [3] M. Baake, P. Kramer, M. Schlottmann, D. Zeidler, Planar patterns with fivefold symmetry as sections of periodic structures in 4-space, Internat. J. Modern Phys. B 4 (15–16) (1990) 2217–2268. [4] N.G. de Bruijn, Algebraic theory of Penrose’s nonperiodic tilings of the plane, I, II, Indag. Math. (N.S.) 43 (1) (1981) 39–52, 53-66. [5] W.A. Deuber, M. Simonovits, V.T. Sós, A note on paradoxical metric spaces, Studia Sci. Math. Hungar. 30 (1–2) (1995) 17–23. [6] M. Duneau, C. Oguey, Displacive transformations and quasicrystalline symmetries, J. Physique 51 (1) (1990) 5–19. [7] F. Gähler, J. Rhyner, Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings, J. Phys. A 19 (2) (1986) 267–277.
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