Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T23:10:44.108Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Steinhaus tiling problem

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Mihail N. Kolountzakis  and
Thomas Wolff
Mihail N. Kolountzakis
Affiliation:
Department of Mathematics, University of Crete, 714 09 Iraklion, Crete, Greece. e-mail: kolount@math.uch.gr
Thomas Wolff
Affiliation:
Department of Mathematics, 253-37 Caltech, Pasadena, CA 91125, USA. e-mail: wolff@cco.caltech.edu
Get access

Abstract

Several results are proved related to a question of Steinhaus: is there a set E⊂ℝ2 such that the image of E under each rigid motion of IR2 contains exactly one lattice point? Assuming measurability, the analogous question in higher dimensions is answered in the negative, and on the known partial results in the two dimensional case are improved on. Also considered is a related problem involving finite sets of rotations.

MSC classification

Secondary: 52A37: Other problems of combinatorial convexity
Type
Research Article
Information
Mathematika , Volume 46 , Issue 2 , December 1999 , pp. 253 - 280
Copyright
Copyright © University College London 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Beck, J.. On a lattice point problem of H. Steinhaus. Studio Sci. Math. Hung., 24 (1989), 263–268.Google Scholar
2. Berndtsson, B.. Zeros of analytic functions of several variables. Ark. Mat., 16 (1978), 251–262.CrossRefGoogle Scholar
3. Croft, H. T.. Three lattice-point problems of Steinhaus. Quart. J. Math., 33 (1982), 71–83.CrossRefGoogle Scholar
4. Ciucu, M.. A remark on sets having the Steinhaus property. Combinatorial, 16 (1996). 321–324.CrossRefGoogle Scholar
5. Hooley, C.. On the intervals between numbers that are sums of two squares. Ada Math., 127 (1971), 280–297.Google Scholar
6. Hormander, L.. The Analysis of Linear Partial Differential Operators, vol. 1, 2nd edition (Springer-Verlag, 1990).Google Scholar
7. Huxley, M. N.. Exponential sums and lattice points, II. Proc. London Math. Soc, 66 (1993). 279–301.CrossRefGoogle Scholar
8. Iwaniec, H. and Mozzochi, C. J.. On the divisor and circle problems. J. Number Theory. 29 (1988), 60–93.CrossRefGoogle Scholar
9. Kolountzakis, M. N.. A new estimate for a problem of Steinhaus. Intern. Math. Res. Notices. 11 (1996), 547–555.CrossRefGoogle Scholar
10. Kolountzakis, M. N.. Multi-lattice tiles. Intern. Math. Res. Notices, 19 (1997), 937 952.CrossRefGoogle Scholar
11. Montgomery, H.. Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics (American Mathematical Society, 1994).CrossRefGoogle Scholar
12. Ronkin, L. I.. Real sets of uniqueness for entire functions of several variables and the completeness of the system of functions ei(λ, x). Sibirsk. Mat. Z, 13 (1972), 638–644.Google Scholar
13. Sogge, C.. Fourier Integrals in Classical Analysis. (Cambridge University Press, 1993).CrossRefGoogle Scholar
14. Stein, E. M., Weiss, G. L.. Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, 1971).Google Scholar