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