Steinhaus tiling problem and integral quadratic forms
HTML articles powered by AMS MathViewer
- by Wai Kiu Chan and R. Daniel Mauldin PDF
- Proc. Amer. Math. Soc. 135 (2007), 337-342 Request permission
Abstract:
A lattice $L$ in $\mathbb {R}^n$ is said to be equivalent to an integral lattice if there exists a real number $r$ such that the dot product of any pair of vectors in $rL$ is an integer. We show that if $n \geq 3$ and $L$ is equivalent to an integral lattice, then there is no measurable Steinhaus set for $L$, a set which no matter how translated and rotated contains exactly one vector in $L$.References
- József Beck, On a lattice-point problem of H. Steinhaus, Studia Sci. Math. Hungar. 24 (1989), no. 2-3, 263–268. MR 1051154
- J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835
- H. T. Croft, Three lattice-point problems of Steinhaus, Quart. J. Math. Oxford Ser. (2) 33 (1982), no. 129, 71–83. MR 689852, DOI 10.1093/qmath/33.1.71
- William Duke and Rainer Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99 (1990), no. 1, 49–57. MR 1029390, DOI 10.1007/BF01234411
- J. S. Hsia and M. I. Icaza, Effective version of Tartakowsky’s theorem, Acta Arith. 89 (1999), no. 3, 235–253. MR 1691853, DOI 10.4064/aa-89-3-235-253
- John S. Hsia, Yoshiyuki Kitaoka, and Martin Kneser, Representations of positive definite quadratic forms, J. Reine Angew. Math. 301 (1978), 132–141. MR 560499, DOI 10.1515/crll.1978.301.132
- Steve Jackson and R. Daniel Mauldin, On a lattice problem of H. Steinhaus, J. Amer. Math. Soc. 15 (2002), no. 4, 817–856. MR 1915820, DOI 10.1090/S0894-0347-02-00400-9
- Steve Jackson and R. Daniel Mauldin, Survey of the Steinhaus tiling problem, Bull. Symbolic Logic 9 (2003), no. 3, 335–361. MR 2005953, DOI 10.2178/bsl/1058448676
- Mihail N. Kolountzakis and Michael Papadimitrakis, The Steinhaus tiling problem and the range of certain quadratic forms, Illinois J. Math. 46 (2002), no. 3, 947–951. MR 1951250
- Mihail N. Kolountzakis and Thomas Wolff, On the Steinhaus tiling problem, Mathematika 46 (1999), no. 2, 253–280. MR 1832620, DOI 10.1112/S0025579300007750
- R. Daniel Mauldin and Andrew Q. Yingst, Comments about the Steinhaus tiling problem, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2071–2079. MR 1963752, DOI 10.1090/S0002-9939-03-07089-8
- O.T. O’Meara, Introduction to quadratic forms, Springer Verlag, New York, 1963.
- Ken Ono and K. Soundararajan, Ramanujan’s ternary quadratic form, Invent. Math. 130 (1997), no. 3, 415–454. MR 1483991, DOI 10.1007/s002220050191
- Arnold E. Ross and Gordon Pall, An extension of a problem of Kloosterman, Amer. J. Math. 68 (1946), 59–65. MR 14378, DOI 10.2307/2371740
- Rainer Schulze-Pillot, On explicit versions of Tartakovski’s theorem, Arch. Math. (Basel) 77 (2001), no. 2, 129–137. MR 1842088, DOI 10.1007/PL00000471
- J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216, DOI 10.1007/978-1-4684-9884-4
- 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.
- G. L. Watson, Integral quadratic forms, Cambridge Tracts in Mathematics and Mathematical Physics, No. 51, Cambridge University Press, New York, 1960. MR 0118704
Additional Information
- Wai Kiu Chan
- Affiliation: Department of Mathematics and Computer Science, Wesleyan University, Middletown, Connecticut 06459
- MR Author ID: 336822
- Email: wkchan@wesleyan.edu
- R. Daniel Mauldin
- Affiliation: Department of Mathematics, Box 311430, University of North Texas, Denton, Texas 76203
- Email: mauldin@unt.edu
- Received by editor(s): August 8, 2005
- Received by editor(s) in revised form: August 29, 2005
- Published electronically: 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 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 337-342
- MSC (2000): Primary 11E12, 11H06, 28A20
- DOI: https://doi.org/10.1090/S0002-9939-06-08479-6
- MathSciNet review: 2255279