Fuglede’s conjecture for a union of two intervals
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- by I. Łaba
- Proc. Amer. Math. Soc. 129 (2001), 2965-2972
- DOI: https://doi.org/10.1090/S0002-9939-01-06035-X
- Published electronically: March 15, 2001
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Abstract:
We prove that a union of two intervals in $\mathbf R$ is a spectral set if and only if it tiles $\mathbf R$ by translations.References
- Ethan M. Coven and Aaron Meyerowitz, Tiling the integers with translates of one finite set, J. Algebra 212 (1999), no. 1, 161–174. MR 1670646, DOI 10.1006/jabr.1998.7628
- 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
- Alex Iosevich, Nets Katz, and Steen Pedersen, Fourier bases and a distance problem of Erdős, Math. Res. Lett. 6 (1999), no. 2, 251–255. MR 1689215, DOI 10.4310/MRL.1999.v6.n2.a13
- A. Iosevich, N. H. Katz, T. Tao: Convex bodies with a point of curvature do not have Fourier bases, Amer. J. Math, to appear.
- A. Iosevich, N. H. Katz, T. Tao: preprint in preparation.
- Alex Iosevich and Steen Pedersen, Spectral and tiling properties of the unit cube, Internat. Math. Res. Notices 16 (1998), 819–828. MR 1643694, DOI 10.1155/S1073792898000506
- Palle E. T. Jørgensen, Spectral theory of finite volume domains in $\textbf {R}^{n}$, Adv. in Math. 44 (1982), no. 2, 105–120. MR 658536, DOI 10.1016/0001-8708(82)90001-9
- Palle E. T. Jorgensen and Steen Pedersen, Spectral theory for Borel sets in $\textbf {R}^n$ of finite measure, J. Funct. Anal. 107 (1992), no. 1, 72–104. MR 1165867, DOI 10.1016/0022-1236(92)90101-N
- P. Jorgensen, S. Pedersen: Spectral pairs in Cartesian coordinates, J. Fourier Anal. Appl. 5 (1999), 285–302.
- M. Kolountzakis: Non-symmetric convex domains have no basis of exponentials, Illinois J. Math. 44 (2000), 542–550.
- M. Kolountzakis: Packing, tiling, orthogonality, and completeness, Bull. London Math. Soc. 32 (2000), 589–599.
- J. C. Lagarias, J. A. Reed, Y. Wang: Orthonormal bases of exponentials for the $n$-cube, Duke Math. J. 103 (2000), 25–37.
- J. C. Lagarias, S. Szabó: Universal spectra and Tijdeman’s conjecture on factorization of cyclic groups, J. Fourier Anal. Appl., to appear.
- Jeffrey C. Lagarias and Yang Wang, Tiling the line with translates of one tile, Invent. Math. 124 (1996), no. 1-3, 341–365. MR 1369421, DOI 10.1007/s002220050056
- Steen Pedersen, Spectral sets whose spectrum is a lattice with a base, J. Funct. Anal. 141 (1996), no. 2, 496–509. MR 1418517, DOI 10.1006/jfan.1996.0139
- Donald J. Newman, Tesselation of integers, J. Number Theory 9 (1977), no. 1, 107–111. MR 429720, DOI 10.1016/0022-314X(77)90054-3
- Steen Pedersen, Spectral theory of commuting selfadjoint partial differential operators, J. Funct. Anal. 73 (1987), no. 1, 122–134. MR 890659, DOI 10.1016/0022-1236(87)90061-9
- Steen Pedersen, Spectral sets whose spectrum is a lattice with a base, J. Funct. Anal. 141 (1996), no. 2, 496–509. MR 1418517, DOI 10.1006/jfan.1996.0139
- S. Pedersen, Y. Wang: Universal spectra, universal tiling sets and the spectral set conjecture, Scand. J. Math., to appear.
- R. Tijdeman, Decomposition of the integers as a direct sum of two subsets, Number theory (Paris, 1992–1993) London Math. Soc. Lecture Note Ser., vol. 215, Cambridge Univ. Press, Cambridge, 1995, pp. 261–276. MR 1345184, DOI 10.1017/CBO9780511661990.016
Bibliographic Information
- I. Łaba
- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- Address at time of publication: Department of Mathematics, University of British Columbia, Vancouver, Canada V6T 1Z2
- Email: ilaba@math.ubc.ca
- Received by editor(s): February 16, 2000
- Published electronically: March 15, 2001
- Communicated by: Christopher D. Sogge
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2965-2972
- MSC (2000): Primary 42A99
- DOI: https://doi.org/10.1090/S0002-9939-01-06035-X
- MathSciNet review: 1840101