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A Danzer set for axis parallel boxes

Authors: David Simmons and Yaar Solomon
Journal: Proc. Amer. Math. Soc. 144 (2016), 2725-2729
MSC (2010): Primary 54H20, 65D18, 68R01
Published electronically: October 21, 2015
MathSciNet review: 3477090
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Abstract | References | Similar Articles | Additional Information

Abstract: We present concrete constructions of discrete sets in $ \mathbb{R}^d$ ($ d\ge 2$) that intersect every aligned box of volume $ 1$ in $ \mathbb{R}^d$, and which have optimal growth rate $ O(T^d)$.

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  • [AS] Noga Alon and Joel H. Spencer, The probabilistic method, 3rd ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2008. With an appendix on the life and work of Paul Erdős. MR 2437651
  • [Bi] Christopher J. Bishop, A set containing rectifiable arcs QC-locally but not QC-globally, Pure Appl. Math. Q. 7 (2011), no. 1, 121–138. MR 2900167,
  • [BW] R. P. Bambah and A. C. Woods, On a problem of Danzer, Pacific J. Math. 37 (1971), 295–301. MR 0303419
  • [vdC] J. G. van der Corput, Verteilungsfunktionen. I. Mitt. (German) Proc. Akad. Wet. Amsterdam 38 (1935), 813-821.
  • [CFG] Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved problems in geometry, Problem Books in Mathematics, Springer-Verlag, New York, 1991. Unsolved Problems in Intuitive Mathematics, II. MR 1107516
  • [Go] T. Gowers, Rough structures and classification, Visions in mathematics, N. Alon, J. Bourgain, A. Connes, M. Gromov, V. Milman (eds.), Birkhäuser (2000), 79-117.
  • [GL] 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
  • [Ha] J. H. Halton, Algorithm 247: Radical-inverse quasi-random point sequence, Cummon. AMC 7, 12, (1964), 701-702.
  • [Jo] Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, N. Y., 1948, pp. 187–204. MR 0030135
  • [Ma] Jiří Matoušek, Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002. MR 1899299
  • [Sk] M. M. Skriganov, Ergodic theory on 𝑆𝐿(𝑛), Diophantine approximations and anomalies in the lattice point problem, Invent. Math. 132 (1998), no. 1, 1–72. MR 1618631,
  • [SW] Y. Solomon and B. Weiss, Dense forests and Danzer sets, preprint 2014,

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Additional Information

David Simmons
Affiliation: Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom

Yaar Solomon
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York

Received by editor(s): January 21, 2015
Received by editor(s) in revised form: July 21, 2015
Published electronically: October 21, 2015
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society

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