Journal of the American Mathematical Society

Author(s): Steve Jackson; R. Daniel Mauldin
Journal: J. Amer. Math. Soc. 15 (2002), 817-856.
MSC (2000): Primary 04A20; Secondary 11H31
Posted: June 13, 2002
Retrieve article in: PDF DVI PostScript
This article is available free of charge

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.: J. Beck, On a lattice point problem of H. Steinhaus, Studia Sci. Math. Hung. 24 (1989), 263-268. MR 91i:11072
2.: H. T. Croft, Three lattice point problems of Steinhaus, Quart. J. Math. Oxford 33 (1982), 71-83. MR 85g:11051
3.: H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Springer-Verlag, New York, 1991. MR 92c:52001
4.: P. Erdos, P.M. Gruber, and J. Hammer, Lattice points, Longman Sci. Tech., Harlow, 1989. MR 90g:11081
5.: P. Erdos, Problems and results in combinatorial geometry, Discrete Geometry and Convexity, 44 (1985), New York Academy of Sciences, 1-10. MR 87g:52001
6.: P. Erdos, S. Jackson, and R. D. Mauldin, On partitions of lines and space, Fund. Math. 145 (1994), 101-119. MR 95k:04003
7.: P. Erdos, S. Jackson, and R. D. Mauldin, On infinite partitions of lines and space, Fund. Math. 152 (1997), 75-95. MR 98b:03066
8.: C. G. Gibson and P. E. Newstead, On the Geometry of the Planar -Bar Mechanism, Acta Applicandae Mathematicae 7 (1986), 113-135. MR 88a:14037
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, 7 Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. MR 91e:70005
11.: M. 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), 559-565, Progr. Math., 139, Birkhäuser Boston, Boston, MA, 1996. MR 97h:11114
12.: M. N. Kolountzakis and T. Wolff, On the Steinhaus tiling problem, Mathematika 46 (1999), no. 2, 253-280. MR 2002c:52024
13.: P. Komjáth, A lattice point problem of Steinhaus, Quart. J. Math. Oxford 43 (1992), 235-241. MR 93c:04003
14.: W. Sierpinski, Sur un problème de H. Steinhaus concernant les ensembles de points sur le plan, Fund. Math. 46 (1958), 191-194. MR 21:851

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 04A20, 11H31

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: 10.1090/S0894-0347-02-00400-9
PII: S 0894-0347(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
Posted: 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
Copyright of article: Copyright 2002, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN: 1088-6834(e) ISSN: 0894-0347(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS