Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Perfectly ordered quasicrystals and the Littlewood conjecture

Authors: Alan Haynes, Henna Koivusalo and James Walton
Journal: Trans. Amer. Math. Soc. 370 (2018), 4975-4992
MSC (2010): Primary 11J13, 52C23
Published electronically: February 8, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Linearly repetitive cut and project sets are mathematical models for perfectly ordered quasicrystals. In a previous paper we presented a characterization of linearly repetitive cut and project sets. In this paper we extend the classical definition of linear repetitivity to try to discover whether or not there is a natural class of cut and project sets which are models for quasicrystals which are better than `perfectly ordered'. In the positive direction, we demonstrate an uncountable collection of such sets (in fact, a collection with large Hausdorff dimension) for every choice of dimension of the physical space. On the other hand, we show that, for many natural versions of the problems under consideration, the existence of these sets turns out to be equivalent to the negation of a well-known open problem in Diophantine approximation, the Littlewood conjecture.

References [Enhancements On Off] (What's this?)

  • [1] Dzmitry Badziahin, Andrew Pollington, and Sanju Velani, On a problem in simultaneous Diophantine approximation: Schmidt's conjecture, Ann. of Math. (2) 174 (2011), no. 3, 1837-1883. MR 2846492,
  • [2] Keith Ball, An elementary introduction to modern convex geometry, Flavors of geometry, Math. Sci. Res. Inst. Publ., vol. 31, Cambridge Univ. Press, Cambridge, 1997, pp. 1-58. MR 1491097,
  • [3] Victor Beresnevich, Badly approximable points on manifolds, Invent. Math. 202 (2015), no. 3, 1199-1240. MR 3425389,
  • [4] Victor Beresnevich, Detta Dickinson, and Sanju Velani, Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc. 179 (2006), no. 846, x+91. MR 2184760,
  • [5] V. Beresnevich, A. Haynes, and S. Velani, The distribution of $ n\alpha $ and multiplicative Diophantine approximation, preprint.
  • [6] Valérie Berthé and Laurent Vuillon, Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences, Discrete Math. 223 (2000), no. 1-3, 27-53. MR 1782038,
  • [7] Adnene Besbes, Michael Boshernitzan, and Daniel Lenz, Delone sets with finite local complexity: linear repetitivity versus positivity of weights, Discrete Comput. Geom. 49 (2013), no. 2, 335-347. MR 3017915,
  • [8] J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957. MR 0087708
  • [9] J. W. S. Cassels and H. P. F. Swinnerton-Dyer, On the product of three homogeneous linear forms and the indefinite ternary quadratic forms, Philos. Trans. Roy. Soc. London. Ser. A. 248 (1955), 73-96. MR 0070653,
  • [10] Manfred Einsiedler, Lior Fishman, and Uri Shapira, Diophantine approximations on fractals, Geom. Funct. Anal. 21 (2011), no. 1, 14-35. MR 2773102,
  • [11] Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. of Math. (2) 164 (2006), no. 2, 513-560. MR 2247967,
  • [12] P. Gallagher, Metric simultaneous diophantine approximation, J. London Math. Soc. 37 (1962), 387-390. MR 0157939,
  • [13] P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, 2nd ed., North-Holland Mathematical Library, vol. 37, North-Holland Publishing Co., Amsterdam, 1987. MR 893813
  • [14] Alan Haynes, Michael Kelly, and Barak Weiss, Equivalence relations on separated nets arising from linear toral flows, Proc. Lond. Math. Soc. (3) 109 (2014), no. 5, 1203-1228. MR 3283615,
  • [15] Alan Haynes, Henna Koivusalo, James Walton, and Lorenzo Sadun, Gaps problems and frequencies of patches in cut and project sets, Math. Proc. Cambridge Philos. Soc. 161 (2016), no. 1, 65-85. MR 3505670,
  • [16] A. Haynes, H. Koivusalo, and J. Walton, A characterization of linearly repetitive cut and project sets, preprint, arXiv:1503.04091.
  • [17] Antoine Julien, Complexity and cohomology for cut-and-projection tilings, Ergodic Theory Dynam. Systems 30 (2010), no. 2, 489-523. MR 2599890,
  • [18] Jeffrey C. Lagarias and Peter A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems 23 (2003), no. 3, 831-867. MR 1992666,
  • [19] Elon Lindenstrauss, Adelic dynamics and arithmetic quantum unique ergodicity, Current developments in mathematics, 2004, Int. Press, Somerville, MA, 2006, pp. 111-139. MR 2459293
  • [20] Kurt Mahler, Ein Übertragungsprinzip für lineare Ungleichungen, Časopis Pěst. Mat. Fys. 68 (1939), 85-92 (German). MR 0001241
  • [21] Andrew D. Pollington and Sanju L. Velani, On a problem in simultaneous Diophantine approximation: Littlewood's conjecture, Acta Math. 185 (2000), no. 2, 287-306. MR 1819996,
  • [22] Lorenzo Sadun, Topology of tiling spaces, University Lecture Series, vol. 46, American Mathematical Society, Providence, RI, 2008. MR 2446623
  • [23] Wolfgang M. Schmidt, Badly approximable systems of linear forms, J. Number Theory 1 (1969), 139-154. MR 0248090,
  • [24] T. Tao, A weakening of the Littlewood conjecture, MathOverflow, (version: 2015-06-18).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11J13, 52C23

Retrieve articles in all journals with MSC (2010): 11J13, 52C23

Additional Information

Alan Haynes
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204

Henna Koivusalo
Affiliation: Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom
Address at time of publication: Department of Mathematics, University of Vienna, Vienna, Austria

James Walton
Affiliation: Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom
Address at time of publication: Department of Mathematical Sciences, University of Durham, Durham, United Kingdom

Received by editor(s): May 13, 2016
Received by editor(s) in revised form: November 3, 2016
Published electronically: February 8, 2018
Additional Notes: This research was supported by EPSRC grants EP/L001462, EP/J00149X, EP/M023540
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society