Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-16T04:01:33.023Z Has data issue: false hasContentIssue false
  • English
  • Français

Survey of the Steinhaus Tiling Problem

Published online by Cambridge University Press:  15 January 2014

Steve Jackson  and
R. Daniel Mauldin
Steve Jackson
Affiliation:
Department of Mathematics, BOX 311430, University of North Texas, Denton, TX 76203, USAE-mail:jackson@unt.edu, E-mail:mauldin@unt.edu
R. Daniel Mauldin
Affiliation:
Department of Mathematics, BOX 311430, University of North Texas, Denton, TX 76203, USAE-mail:jackson@unt.edu, E-mail:mauldin@unt.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We survey some results and problems arising from a classic problem of Steinhaus: Is there a subset S of ℝ2 such that each isometric copy of ℤ2 (the lattice points in the plane) meets S in exactly one point.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 9 , Issue 3 , September 2003 , pp. 335 - 361
Copyright
Copyright © Association for Symbolic Logic 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Beck, J., On a lattice point problem of H. Steinhaus, Studia Scientiarum Mathematicarum Hungarica, vol. 24 (1989), pp. 263–268.Google Scholar
[2] Ciucu, M., A remark on sets having the Steinhaus property, Combinatorica, vol. 16 (1996), pp. 321–324.CrossRefGoogle Scholar
[3] Croft, H. T., Three lattice point problems of Steinhaus, The Quarterly Journal of Mathematics. Oxford, vol. 33 (1982), pp. 71–83.CrossRefGoogle Scholar
[4] Croft, H. T., Falconer, K. J., and Guy, R. K., Unsolved problems in geometry, Springer-Verlag, New York, 1991.CrossRefGoogle Scholar
[5] Dickson, Leonard Eugene, Modern elementary theory of numbers, The University of Chicago Press, Chicago, 1939, pp. 109–113.Google Scholar
[6] Erdős, P., Problems and results in combinatorial geometry, Discrete geometry and convexity, vol. 44, New York Academy of Sciences, 1985, pp. 1–10.Google Scholar
[7] Erdős, P., Gruber, P.M., and Hammer, J., Lattice points, Longman Sci. Tech., Harlow, 1989.Google Scholar
[8] Erdős, P., Jackson, S., and Mauldin, R.D., On partitions of lines and space, Fundamenta Mathematicae, vol. 145 (1994), pp. 101–119.Google Scholar
[9] Erdős, P., Jackson, S., and Mauldin, R.D., On infinite partitions of lines and space, Fundamenta Mathematicae, vol. 152 (1997), pp. 75–95.Google Scholar
[10] Gibson, C. G. and Newstead, P. E., On the geometry of the planar 4-bar mechanism, Acta Applicandae Mathematicae, vol. 7 (1986), pp. 113–135.CrossRefGoogle Scholar
[11] Hunt, K. H., Kinematic geometry of mechanisms, Oxford Engineering Science Series, vol. 7, Oxford Science Publications, The Clarendon Press, Oxford University press, New York, 1990.Google Scholar
[12] Huxley, M., Exponential sums and lattice points, Proceedings of the London Mathematical Society, vol. 66 (1993), no. 3, pp. 279–301.CrossRefGoogle Scholar
[13] Jackson, S. and Mauldin, R. D., On a lattice problem of H. Steinhaus, Journal of the American Mathematical Society, vol. 15 (2002), pp. 817–856.CrossRefGoogle Scholar
[14] Jackson, S. and Mauldin, R. D., Sets meeting isometric copies of a lattice in exactly one point, Proceedings of the National Academy of Sciences of the United States of America, vol. 99 (2002), pp. 15883–15887.Google Scholar
[15] Kolountzakis, M.N., A new estimate for a problem of Steinhaus, International Mathematics Research Notices, vol. 11 (1996), pp. 547–555.CrossRefGoogle Scholar
[16] Kolountzakis, M.N., A problem of Steinhaus: Can all placements of a planar set contain exactly one lattice point?, Analytic number theory, vol. 2 (Allerton Park, IL, 1995), Progress in Mathematics, vol. 139, Birkhäuser Boston, Boston, MA, 1996, pp. 559–565.Google Scholar
[17] Kolountzakis, M.N., Multi-lattice tiles, International Mathematics Research Notices, vol. 19 (1997), pp. 937–952.CrossRefGoogle Scholar
[18] Kolountzakis, M.N., The study of translational tiling with Fourier analysis, Proceedings of the Milano conference on Fourier analysis and convexity, to appear.Google Scholar
[19] Kolountzakis, M. N. and Papadimitrakis, M., The Steinhaus tiling problem and the range of certain quadratic forms, Illinois Journal of Mathematics, vol. 46 (2002), pp. 947–951.Google Scholar
[20] Kolountzakis, M. N. and Wolff, T., On the Steinhaus tiling problem, Mathematika, vol. 46 (1999), pp. 253–280.Google Scholar
[21] Komjáth, P., A lattice point problem of Steinhaus, The Quarterly Journal of Mathematics. Oxford, vol. 43 (1992), pp. 235–241.Google Scholar
[22] Mauldin, R. D., On sets which meet each line in exactly two points, The Bulletin of the London Mathematical Society, vol. 30 (1998), pp. 397–403.Google Scholar
[23] Mauldin, R. D. and Yingst, A. Q., Comments on the Steinhaus tiling problem, Proceedings of the American Mathematical Society, vol. 131 (2003), pp. 2071–2079.Google Scholar
[24] Shelah, Saharon, Can you take Solovay's inaccessible away?, Israel Journal of Mathematics, vol. 48 (1984), pp. 1–47.Google Scholar
[25] Sierpiński, W., Sur un problème de H. Steinhaus concernant les ensembles de points sur le plan, Fundamenta Mathematicae, vol. 46 (1958), pp. 191–194.CrossRefGoogle Scholar
[26] Solovay, Robert M., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, (1970), pp. 1–56.Google Scholar
[27] Srivastava, S. M. and Thangadurai, R., Steinhaus sets are disconnected, preprint.Google Scholar
[28] Stein, E. M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, New Jersey, 1971.Google Scholar