Multiple lattice tiles and Riesz bases of exponentials
HTML articles powered by AMS MathViewer
- by Mihail N. Kolountzakis PDF
- Proc. Amer. Math. Soc. 143 (2015), 741-747 Request permission
Abstract:
Suppose $\Omega \subseteq \mathbb {R}^d$ is a bounded and measurable set and $\Lambda \subseteq \mathbb {R}^d$ is a lattice. Suppose also that $\Omega$ tiles multiply, at level $k$, when translated at the locations $\Lambda$. This means that the $\Lambda$-translates of $\Omega$ cover almost every point of $\mathbb {R}^d$ exactly $k$ times. We show here that there is a set of exponentials $\exp (2\pi i t\cdot x)$, $t\in T$, where $T$ is some countable subset of $\mathbb {R}^d$, which forms a Riesz basis of $L^2(\Omega )$. This result was recently proved by Grepstad and Lev under the extra assumption that $\Omega$ has boundary of measure $0$, using methods from the theory of quasicrystals. Our approach is rather more elementary and is based almost entirely on linear algebra. The set of frequencies $T$ turns out to be a finite union of shifted copies of the dual lattice $\Lambda ^*$. It can be chosen knowing only $\Lambda$ and $k$ and is the same for all $\Omega$ that tile multiply with $\Lambda$.References
- U. Bolle, On multiple tiles in $E^2$, Intuitive geometry (Szeged, 1991) Colloq. Math. Soc. János Bolyai, vol. 63, North-Holland, Amsterdam, 1994, pp. 39–43. MR 1383609
- Bálint Farkas, Máté Matolcsi, and Péter Móra, On Fuglede’s conjecture and the existence of universal spectra, J. Fourier Anal. Appl. 12 (2006), no. 5, 483–494. MR 2267631, DOI 10.1007/s00041-005-5069-7
- Bálint Farkas and Szilárd Gy. Révész, Tiles with no spectra in dimension 4, Math. Scand. 98 (2006), no. 1, 44–52. MR 2221543, DOI 10.7146/math.scand.a-14982
- Bent Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Functional Analysis 16 (1974), 101–121. MR 0470754, DOI 10.1016/0022-1236(74)90072-x
- Nick Gravin, Sinai Robins, and Dmitry Shiryaev, Translational tilings by a polytope, with multiplicity, Combinatorica 32 (2012), no. 6, 629–649. MR 3063154, DOI 10.1007/s00493-012-2860-3
- Sigrid Grepstad and Nir Lev, Multi-tiling and Riesz bases, arXiv preprint arXiv:1212.4679 (2012).
- Alex Iosevich, Nets Hawk Katz, and Terry Tao, Convex bodies with a point of curvature do not have Fourier bases, Amer. J. Math. 123 (2001), no. 1, 115–120. MR 1827279
- Alex Iosevich, Nets Katz, and Terence Tao, The Fuglede spectral conjecture holds for convex planar domains, Math. Res. Lett. 10 (2003), no. 5-6, 559–569. MR 2024715, DOI 10.4310/MRL.2003.v10.n5.a1
- Mihail N. Kolountzakis and Máté Matolcsi, Complex Hadamard matrices and the spectral set conjecture, Collect. Math. Vol. Extra (2006), 281–291. MR 2264214
- M. N. Kolountzakis, On the structure of multiple translational tilings by polygonal regions, Discrete Comput. Geom. 23 (2000), no. 4, 537–553. MR 1753701, DOI 10.1007/s004540010014
- Mihail N. Kolountzakis, The study of translational tiling with Fourier analysis, Fourier analysis and convexity, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2004, pp. 131–187. MR 2087242
- Mihail N. Kolountzakis and Máté Matolcsi, Tiles with no spectra, Forum Math. 18 (2006), no. 3, 519–528. MR 2237932, DOI 10.1515/FORUM.2006.026
- Gady Kozma and Shahaf Nitzan, Combining Riesz bases, arXiv preprint arXiv:1210.6383 (2012).
- Yurii I. Lyubarskii and Alexander Rashkovskii, Complete interpolating sequences for Fourier transforms supported by convex symmetric polygons, Ark. Mat. 38 (2000), no. 1, 139–170. MR 1749363, DOI 10.1007/BF02384495
- Yurii I. Lyubarskii and Kristian Seip, Sampling and interpolating sequences for multiband-limited functions and exponential bases on disconnected sets, J. Fourier Anal. Appl. 3 (1997), no. 5, 597–615. Dedicated to the memory of Richard J. Duffin. MR 1491937, DOI 10.1007/BF02648887
- Jordi Marzo, Riesz basis of exponentials for a union of cubes in ${\mathbb R}^d$, arXiv preprint math/0601288 (2006).
- Basarab Matei and Yves Meyer, Quasicrystals are sets of stable sampling, C. R. Math. Acad. Sci. Paris 346 (2008), no. 23-24, 1235–1238 (English, with English and French summaries). MR 2473299, DOI 10.1016/j.crma.2008.10.006
- Basarab Matei and Yves Meyer, Simple quasicrystals are sets of stable sampling, Complex Var. Elliptic Equ. 55 (2010), no. 8-10, 947–964. MR 2674875, DOI 10.1080/17476930903394689
- Alexander Olevskii and Alexander Ulanovskii, On multi-dimensional sampling and interpolation, Anal. Math. Phys. 2 (2012), no. 2, 149–170. MR 2917231, DOI 10.1007/s13324-012-0027-4
- Terence Tao, Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), no. 2-3, 251–258. MR 2067470, DOI 10.4310/MRL.2004.v11.n2.a8
Additional Information
- Mihail N. Kolountzakis
- Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, GR-700 13, Heraklion, Crete, Greece
- Email: kolount@math.uoc.gr
- Received by editor(s): May 12, 2013
- Published electronically: October 8, 2014
- Additional Notes: The author was supported in part by grant No 3803 from the University of Crete.
- Communicated by: Alexander Iosevich
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 741-747
- MSC (2010): Primary 42B99
- DOI: https://doi.org/10.1090/S0002-9939-2014-12310-0
- MathSciNet review: 3283660