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

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

Published electronically:
February 13, 2003

MathSciNet review:
1963752

Full-text PDF

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.**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. 33 (1982), 71-82. MR**85g:11051****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.**M. N. Kolountzakis and T. Wolff, On the Steinhaus tiling problem, Mathematika 46 (1999), 253-280. MR**2002c:52024****6.**Leonard Eugene Dickson, Modern Elementary Theory of Numbers, The University of Chicago Press, Chicago, 1939 (see pp. 109-113). MR**1:65a**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
28A20,
11K31,
60F20

Retrieve articles in all journals with MSC (2000): 28A20, 11K31, 60F20

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:
https://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