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On the discrete
Fuglede and Pompeiu problems
Gergely Kiss, Romanos Diogenes Malikiosis, Gábor Somlai and Máté Vizer |
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Vol. 13 (2020), No. 3, 765–788
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Abstract |
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We investigate the discrete Fuglede conjecture and the Pompeiu problem on finite abelian groups and develop a strong connection between the two problems. We give a geometric condition under which a multiset of a finite abelian group has the discrete Pompeiu property. Using this description and the revealed connection we prove that Fuglede’s conjecture holds for , where and are different primes. In particular, we show that every spectral subset of tiles the group. Further, using our combinatorial methods we give a simple proof for the statement that Fuglede’s conjecture holds for . |
Keywords
Pompeiu problem, spectral set conjecture, tiling, vanishing
sums of roots of unity
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Mathematical Subject Classification 2010
Primary: 39B32, 43A46
Secondary: 13F20, 20K01
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Milestones
Received: 10 July 2018
Revised: 2 March 2019
Accepted: 8 April 2019
Published: 15 April 2020
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Authors |
| Gergely Kiss | |
| Faculty of Science Mathematics Research Unit University of Luxembourg Luxembourg Luxembourg |
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| Romanos Diogenes Malikiosis | |
| Department of Mathematics Aristotle University of Thessaloniki Thessaloniki Greece |
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| Gábor Somlai | |
| MTA-ELTE Geometric and Algebraic
Combinatorics Research Group NKFIH 115799 and Faculty of Science, Institute of Mathematics Eötvös Loránd University Budapest Hungary |
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| Máté Vizer | |
| Alfréd Rényi Institute of
Mathematics Hungarian Academy of Sciences Budapest Hungary |
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