On a lattice problem of H. Steinhaus

Jackson, Steve; Mauldin, R.

doi:10.1090/S0894-0347-02-00400-9

On a lattice problem of H. Steinhaus

Authors: Steve Jackson and R. Daniel Mauldin
Journal: J. Amer. Math. Soc. 15 (2002), 817-856
MSC (2000): Primary 04A20; Secondary 11H31
DOI: https://doi.org/10.1090/S0894-0347-02-00400-9
Published electronically: June 13, 2002
MathSciNet review: 1915820
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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that there is a subset of $\mathbb{R} ^2$ such that each isometric copy of $\mathbb{Z} ^2$ (the lattice points in the plane) meets in exactly one point. This provides a positive answer to a problem of H. Steinhaus.

1. József Beck, On a lattice-point problem of H. Steinhaus, Studia Sci. Math. Hungar. 24 (1989), no. 2-3, 263–268. MR 1051154
2. H. T. Croft, Three lattice-point problems of Steinhaus, Quart. J. Math. Oxford Ser. (2) 33 (1982), no. 129, 71–83. MR 689852, https://doi.org/10.1093/qmath/33.1.71
3. 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
4. P. Erdös, P. M. Gruber, and J. Hammer, Lattice points, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 39, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 1003606
5. P. Erdős, Problems and results in combinatorial geometry, Discrete geometry and convexity (New York, 1982) Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, 1985, pp. 1–11. MR 809186, https://doi.org/10.1111/j.1749-6632.1985.tb14533.x
6. P. Erdős, Steve Jackson, and R. Daniel Mauldin, On partitions of lines and space, Fund. Math. 145 (1994), no. 2, 101–119. MR 1297399
7. P. Erdős, Steve Jackson, and R. Daniel Mauldin, On infinite partitions of lines and space, Fund. Math. 152 (1997), no. 1, 75–95. MR 1434378
8. C. G. Gibson and P. E. Newstead, On the geometry of the planar 4-bar mechanism, Acta Appl. Math. 7 (1986), no. 2, 113–135. MR 860287, https://doi.org/10.1007/BF00051348
9. S. Jackson and R. D. Mauldin, Sets meeting isometric copies of a lattice in exactly one point, Proceedings National Academy Sciences, submitted.
10. K. H. Hunt, Kinematic geometry of mechanisms, Oxford Engineering Science Series, vol. 7, The Clarendon Press, Oxford University Press, New York, 1990. Corrected reprint of the 1978 original; Oxford Science Publications. MR 1070564
11. Mihail N. Kolountzakis, 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) Progr. Math., vol. 139, Birkhäuser Boston, Boston, MA, 1996, pp. 559–565. MR 1409378
12. Mihail N. Kolountzakis and Thomas Wolff, On the Steinhaus tiling problem, Mathematika 46 (1999), no. 2, 253–280. MR 1832620, https://doi.org/10.1112/S0025579300007750
13. Péter Komjáth, A lattice-point problem of Steinhaus, Quart. J. Math. Oxford Ser. (2) 43 (1992), no. 170, 235–241. MR 1164626, https://doi.org/10.1093/qmath/43.2.235
14. W. Sierpiński, Sur un problème de H. Steinhaus concernant les ensembles de points sur le plan, Fund. Math. 46 (1959), 191–194 (French). MR 102056, https://doi.org/10.4064/fm-46-2-191-194

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Additional Information

Steve Jackson
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
Email: jackson@unt.edu

R. Daniel Mauldin
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
Email: mauldin@unt.edu

DOI: https://doi.org/10.1090/S0894-0347-02-00400-9
Keywords: Lattice points, Steinhaus problem, four-bar linkage
Received by editor(s): February 14, 2001
Received by editor(s) in revised form: October 29, 2001
Published electronically: June 13, 2002
Additional Notes: The first author’s research was supported by NSF Grant DMS-0097181.
The second author’s research was supported by NSF Grant DMS-9801583
Article copyright: © Copyright 2002 American Mathematical Society