Comments about the Steinhaus tiling problem

Authors:
R. Daniel Mauldin and Andrew Q. Yingst

Journal:
Proc. Amer. Math. Soc. **131** (2003), 2071-2079

MSC (2000):
Primary 28A20; Secondary 11K31, 60F20

Published electronically:
February 13, 2003

MathSciNet review:
1963752

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Abstract | References | Similar Articles | Additional Information

Abstract: Recently, using Fourier transform methods, it was shown that there is no measurable Steinhaus set in , a set which no matter how translated and rotated contains exactly one integer lattice point. Here, we show that this argument cannot generalize to any lattice and, on the other hand, give some lattices to which this method applies. We also show there is no measurable Steinhaus set for a special honeycomb lattice, the standard tetrahedral lattice in

**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**, 10.1093/qmath/33.1.71**3.**M. N. Kolountzakis and M. Papadimitrakis, The Steinhaus tiling problem and the range of certain quadratic forms, to appear.**4.**Steve Jackson and R. Daniel Mauldin, On a lattice problem of Steinhaus, Jour. Amer. Math. Soc. 15 (2002), 817-856.**5.**Mihail N. Kolountzakis and Thomas Wolff,*On the Steinhaus tiling problem*, Mathematika**46**(1999), no. 2, 253–280. MR**1832620**, 10.1112/S0025579300007750**6.**Leonard Eugene Dickson,*Modern Elementary Theory of Numbers*, University of Chicago Press, Chicago, 1939. MR**0000387**

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Additional Information

**R. Daniel Mauldin**

Affiliation:
Department of Mathematics, Box 311430, University of North Texas, Denton, Texas 76203

Email:
mauldin@unt.edu

**Andrew Q. Yingst**

Affiliation:
Department of Mathematics, Box 311430, University of North Texas, Denton, Texas 76203

Email:
andyq@unt.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-03-07089-8

Keywords:
Method of descent,
Fourier transform,
lattice,
quadratic form

Received by editor(s):
October 15, 2001

Published electronically:
February 13, 2003

Additional Notes:
Both authors were supported in part by NSF Grant DMS 0100078

Communicated by:
David Preiss

Article copyright:
© Copyright 2003
American Mathematical Society