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ISSN 1088-6826(e) ISSN 0002-9939(p)

Steinhaus tiling problem and integral quadratic forms

Author(s): Wai Kiu Chan; R. Daniel Mauldin
Journal: Proc. Amer. Math. Soc. 135 (2007), 337-342.
MSC (2000): Primary 11E12, 11H06, 28A20
Posted: August 4, 2006
MathSciNet review: 2255279
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Abstract: A lattice in $\mathbb{R}^n$ is said to be equivalent to an integral lattice if there exists a real number such that the dot product of any pair of vectors in is an integer. We show that if $n \geq 3$ and is equivalent to an integral lattice, then there is no measurable Steinhaus set for , a set which no matter how translated and rotated contains exactly one vector in .

References:

1.: J. Beck, On a lattice point problem of H. Steinhaus, Studia Sci. Math. Hung. 24 (1989), 263-268. MR 1051154 (91i:11072)
2.: J. W. S. Cassels, Rational quadratic forms, Academic Press, 1978. MR 0522835 (80m:10019)
3.: H. T. Croft, Three lattice-point problems of Steinhaus, Quart. J. Math. 33 (1982), 71-82. MR 0689852 (85g:11051)
4.: W. Duke and R. Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99 (1990), 49-57. MR 1029390 (90m:11051)
5.: J. S. Hsia and M. I. Icaza, Effective version of Tartakowky's theorem, Acta Arith. 89 (1999), 235-253. MR 1691853 (2000h:11031)
6.: J.S. Hsia, Y. Kitaoka, and M. Kneser, Representation by positive quadratic forms, J. Reine Angew. Math. 301 (1977), 132-141. MR 0560499 (58:27758)
7.: S. Jackson and R. D. Mauldin, On a lattic problem of Steinhaus, J. Amer. Math. Soc. 15 (2002), 817-856. MR 1915820 (2004b:52026)
8.: S. Jackson and R. D. Mauldin, Survey of the Steinhaus tiling problem, Bull. Symbolic Logic 9 (2003) no. 3, 335-361. MR 2005953 (2004k:28011)
9.: M. N. Kolountzakis and M. Papadimitrakis, The Steinhaus tiling problem and the range of certain quadratic forms, Illinois J. Math. 46 (2002), 947-951. MR 1951250 (2003m:52026)
10.: M. N. Kolountzakis and T. Wolff, On the Steinhaus tiling problem, Mathematika 46 (1999), 253-280. MR 1832620 (2002c:52024)
11.: R. D. Mauldin and A. Yingst, Comments about the Steinhaus tiling problem, Proc. Amer. Math. Soc. 131 (2003), 2071-2079. MR 1963752 (2003k:28001)
12.: O.T. O'Meara, Introduction to quadratic forms, Springer Verlag, New York, 1963.
13.: K. Ono and K. Soundararajan, Ramanujan's ternary quadratic forms, Invent. Math. 130 (1997), 415-454. MR 1483991 (99b:11036)
14.: A. Ross and G. Pall, An extension of a problem of Kloosterman, Amer. J. Math. 68 (1946), 59-65. MR 0014378 (7:275b)
15.: R. Schulze-Pillot, On explicit versions of Tartakowsky's theorem, Arch. Math. (Basel) 77 (2001), no. 2, 129-137. MR 1842088 (2002i:11036)
16.: J. P. Serre, A Course in arithmetic, Springer-Verlag, 1973. MR 0344216 (49:8956)
17.: W. Tartakowsky, Die Gesamtheit der Zahlen, die durch eine quadratische form $F(x_1, x_2, \ldots, x_s), (s \geq 4)$ darstellbar sind., Izv. Akad. Nauk SSSR (1929), 111-122, 165-196.
18.: G. L. Watson, Intgeral quadratic forms, Cambridge University Press, 1960. MR 0118704 (22:9475)

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Additional Information:

Wai Kiu Chan
Affiliation: Department of Mathematics and Computer Science, Wesleyan University, Middletown, Connecticut 06459
Email: wkchan@wesleyan.edu

R. Daniel Mauldin
Affiliation: Department of Mathematics, Box 311430, University of North Texas, Denton, Texas 76203
Email: mauldin@unt.edu

DOI: 10.1090/S0002-9939-06-08479-6
PII: S 0002-9939(06)08479-6
Keywords: Representations by quadratic forms, Steinhaus tiling problem
Received by editor(s): August 8, 2005
Received by editor(s) in revised form: August 29, 2005
Posted: August 4, 2006
Additional Notes: The research of the first author was partially supported by NSF grant DMS-0138524
The second author was supported in part by NSF grant DMS-0400481
Communicated by: Ken Ono
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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