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March  2022, 42(3): 1403-1414. doi: 10.3934/dcds.2021157

Number of bounded distance equivalence classes in hulls of repetitive Delone sets

Dirk Frettlöh 1, , Alexey Garber 2,, and  Lorenzo Sadun 3,

1. 

Technische Fakultät, Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany
2. 

School of Mathematical & Statistical Sciences, The University of Texas Rio Grande Valley, 1 West University Blvd., Brownsville, TX 78520, USA
3. 

Department of Mathematics, University of Texas, 2515 Speedway, PMA 8.100, Austin, TX 78712, USA

* Corresponding author: Alexey Garber

Received  February 2021 Revised  September 2021 Published  March 2022 Early access  November 2021

Fund Project: A.G. is partially supported by the Alexander von Humboldt Foundation

Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many.

Keywords: Delone set, finite local complexity, topological hull, bounded distance equivalence, aperiodic order, aperiodic tilings.
Mathematics Subject Classification: Primary: 37B51, 52C23.
Citation: Dirk Frettlöh, Alexey Garber, Lorenzo Sadun. Number of bounded distance equivalence classes in hulls of repetitive Delone sets. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1403-1414. doi: 10.3934/dcds.2021157
References:
[1]

J. Aliste-Prieto, D. Coronel and J. M. Gambaudo, Linearly repetitive Delone sets are rectifiable, Ann. Inst. H. Poincaré Anal. Non Lineáire, 30 (2013), 275-290.  doi: 10.1016/j.anihpc.2012.07.006.

[2]

J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated $C^*$-algebras, Ergodic Theory Dynam. Systems, 18 (1998), 509-537.  doi: 10.1017/S0143385798100457.

[3] M. Baake and U. Grimm, Aperiodic Order, Cambridge Univ. Press, Cambridge, 2013. 
[4]

K. Bezdek, From the Kneser–Poulsen conjecture to ball–polyhedra, European J. Combin., 29 (2008), 1820-1830.  doi: 10.1016/j.ejc.2008.01.011.

[5]

B. Csikós, A Schläfli–type formula for polytopes with curved faces and its application to the Kneser–Poulsen conjecture, Monatsh. Math, 147 (2006), 273-292.  doi: 10.1007/s00605-005-0363-7.

[6]

W. A. Deuber, M. Simonovits and V. T. Sós, A note on paradoxical metric spaces, Stud. Sci. Math. Hungar., 30 (1995), 17-23. 

[7]

M. Duneau and C. Oguey, Bounded interpolation between lattices, J. Phys. A: Math. Gen., 24 (1991), 461-475.  doi: 10.1088/0305-4470/24/2/019.

[8]

M. Duneau and C. Oguey, Displacive transformations and quasicrystalline symmetries, J. Phys. France, 51 (1990), 5-19.  doi: 10.1051/jphys:019900051010500.

[9]

N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $ \mathbb{R}^d$, Geom. Dedicata, 171 (2014), 149-186.  doi: 10.1007/s10711-013-9893-7.

[10]

D. Frettlöh and A. Garber, Pisot substitution sequences, one dimensional cut–and–project sets and bounded remainder sets with fractal boundary, Indag. Math., 29 (2018), 1114-1130.  doi: 10.1016/j.indag.2018.05.012.

[11]

D. Frettlöh, Y. Smilansky and Y. Solomon, Bounded displacement non-equivalence in substitution tilings, J. Combin. Theory Ser. A, 177 (2021), 105326.  doi: 10.1016/j.jcta.2020.105326.

[12]

A. I. Garber, On equivalence classes of separated nets (in Russian), Modeling and Analysis of Information Systems, 16 (2009), 109-118. 

[13]

S. Grepstad and N. Lev, Sets of bounded discrepancy for multi–dimensional irrational rotation, Geom. Funct. Anal., 25 (2015), 87-133.  doi: 10.1007/s00039-015-0313-z.

[14]

A. Haynes, Equivalence classes of codimension one cut–and–project nets, Ergodic Theory Dyn. Syst., 36 (2016), 816-831.  doi: 10.1017/etds.2014.90.

[15]

A. Haynes and H. Koivusalo, Constructing bounded remainder sets and cut–and–project sets which are bounded distance to lattices, Israel J. Math., 212 (2016), 189-201.  doi: 10.1007/s11856-016-1283-z.

[16]

A. Haynes, M. Kelly and H. Koivusalo, Constructing bounded remainder sets and cut–and–project sets which are bounded distance to lattices Ⅱ, Indag. Math., 28 (2017), 138-144.  doi: 10.1016/j.indag.2016.11.010.

[17]

C. Holton and L. Zamboni, Geometric realization of substitutions, Bull. Soc. Math. France, 126 (1998), 149-179.  doi: 10.24033/bsmf.2324.

[18]

J. Kellendonk and I. F. T. Putnam, $C^*$-algebras, and $K$-theory, Directions in Mathematical Quasicrystals, CRM Monogr. Ser., 13, Amer. Math. Soc., Providence, RI, 2000,177–206.

[19]

M. Laczkovich, Uniformly spread discrete sets in $ \mathbb{R}^d$, J. London Math. Soc., 46 (1992), 39-57.  doi: 10.1112/jlms/s2-46.1.39.

[20]

J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems, 23 (2003), 831-867.  doi: 10.1017/S0143385702001566.

[21]

R. Rado, Factorization of even graphs, Quart. J. Math. Oxford, 20 (1949), 95-104.  doi: 10.1093/qmath/os-20.1.95.

[22]

L. Sadun, Topology of Tiling Spaces, University Lecture Series, 46, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/ulect/046.

[23]

Y. Smilansky and Y. Solomon, A dichotomy for bounded displacement and Chabauty-Fell convergence of discrete sets, arXiv: 2011.00106, to appear In Erg. Th. and Dyn. Syst..

[24]

Y. Solomon, Substitution tilings and separated nets with similarities to the integer lattice, Israel J. Math., 181 (2011), 445-460.  doi: 10.1007/s11856-011-0018-4.

[25]

Y. Solomon, A simple condition for bounded displacement, J. Math. Anal. Appl., 414 (2014), 134-148.  doi: 10.1016/j.jmaa.2013.12.050.

[26]

Y. Solomon, Continuously many bounded displacement non-equivalences in substitution tiling spaces, J. Math. Anal. Appl., 492 (2020), 124426.  doi: 10.1016/j.jmaa.2020.124426.

show all references

References:
[1]

J. Aliste-Prieto, D. Coronel and J. M. Gambaudo, Linearly repetitive Delone sets are rectifiable, Ann. Inst. H. Poincaré Anal. Non Lineáire, 30 (2013), 275-290.  doi: 10.1016/j.anihpc.2012.07.006.

[2]

J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated $C^*$-algebras, Ergodic Theory Dynam. Systems, 18 (1998), 509-537.  doi: 10.1017/S0143385798100457.

[3] M. Baake and U. Grimm, Aperiodic Order, Cambridge Univ. Press, Cambridge, 2013. 
[4]

K. Bezdek, From the Kneser–Poulsen conjecture to ball–polyhedra, European J. Combin., 29 (2008), 1820-1830.  doi: 10.1016/j.ejc.2008.01.011.

[5]

B. Csikós, A Schläfli–type formula for polytopes with curved faces and its application to the Kneser–Poulsen conjecture, Monatsh. Math, 147 (2006), 273-292.  doi: 10.1007/s00605-005-0363-7.

[6]

W. A. Deuber, M. Simonovits and V. T. Sós, A note on paradoxical metric spaces, Stud. Sci. Math. Hungar., 30 (1995), 17-23. 

[7]

M. Duneau and C. Oguey, Bounded interpolation between lattices, J. Phys. A: Math. Gen., 24 (1991), 461-475.  doi: 10.1088/0305-4470/24/2/019.

[8]

M. Duneau and C. Oguey, Displacive transformations and quasicrystalline symmetries, J. Phys. France, 51 (1990), 5-19.  doi: 10.1051/jphys:019900051010500.

[9]

N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $ \mathbb{R}^d$, Geom. Dedicata, 171 (2014), 149-186.  doi: 10.1007/s10711-013-9893-7.

[10]

D. Frettlöh and A. Garber, Pisot substitution sequences, one dimensional cut–and–project sets and bounded remainder sets with fractal boundary, Indag. Math., 29 (2018), 1114-1130.  doi: 10.1016/j.indag.2018.05.012.

[11]

D. Frettlöh, Y. Smilansky and Y. Solomon, Bounded displacement non-equivalence in substitution tilings, J. Combin. Theory Ser. A, 177 (2021), 105326.  doi: 10.1016/j.jcta.2020.105326.

[12]

A. I. Garber, On equivalence classes of separated nets (in Russian), Modeling and Analysis of Information Systems, 16 (2009), 109-118. 

[13]

S. Grepstad and N. Lev, Sets of bounded discrepancy for multi–dimensional irrational rotation, Geom. Funct. Anal., 25 (2015), 87-133.  doi: 10.1007/s00039-015-0313-z.

[14]

A. Haynes, Equivalence classes of codimension one cut–and–project nets, Ergodic Theory Dyn. Syst., 36 (2016), 816-831.  doi: 10.1017/etds.2014.90.

[15]

A. Haynes and H. Koivusalo, Constructing bounded remainder sets and cut–and–project sets which are bounded distance to lattices, Israel J. Math., 212 (2016), 189-201.  doi: 10.1007/s11856-016-1283-z.

[16]

A. Haynes, M. Kelly and H. Koivusalo, Constructing bounded remainder sets and cut–and–project sets which are bounded distance to lattices Ⅱ, Indag. Math., 28 (2017), 138-144.  doi: 10.1016/j.indag.2016.11.010.

[17]

C. Holton and L. Zamboni, Geometric realization of substitutions, Bull. Soc. Math. France, 126 (1998), 149-179.  doi: 10.24033/bsmf.2324.

[18]

J. Kellendonk and I. F. T. Putnam, $C^*$-algebras, and $K$-theory, Directions in Mathematical Quasicrystals, CRM Monogr. Ser., 13, Amer. Math. Soc., Providence, RI, 2000,177–206.

[19]

M. Laczkovich, Uniformly spread discrete sets in $ \mathbb{R}^d$, J. London Math. Soc., 46 (1992), 39-57.  doi: 10.1112/jlms/s2-46.1.39.

[20]

J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems, 23 (2003), 831-867.  doi: 10.1017/S0143385702001566.

[21]

R. Rado, Factorization of even graphs, Quart. J. Math. Oxford, 20 (1949), 95-104.  doi: 10.1093/qmath/os-20.1.95.

[22]

L. Sadun, Topology of Tiling Spaces, University Lecture Series, 46, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/ulect/046.

[23]

Y. Smilansky and Y. Solomon, A dichotomy for bounded displacement and Chabauty-Fell convergence of discrete sets, arXiv: 2011.00106, to appear In Erg. Th. and Dyn. Syst..

[24]

Y. Solomon, Substitution tilings and separated nets with similarities to the integer lattice, Israel J. Math., 181 (2011), 445-460.  doi: 10.1007/s11856-011-0018-4.

[25]

Y. Solomon, A simple condition for bounded displacement, J. Math. Anal. Appl., 414 (2014), 134-148.  doi: 10.1016/j.jmaa.2013.12.050.

[26]

Y. Solomon, Continuously many bounded displacement non-equivalences in substitution tiling spaces, J. Math. Anal. Appl., 492 (2020), 124426.  doi: 10.1016/j.jmaa.2020.124426.

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