Vol. 13, No. 3, 2020

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On the discrete Fuglede and Pompeiu problems

Gergely Kiss, Romanos Diogenes Malikiosis, Gábor Somlai and Máté Vizer

Vol. 13 (2020), No. 3, 765–788
DOI: 10.2140/apde.2020.13.765
Abstract

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 pnq2, where p and q are different primes. In particular, we show that every spectral subset of pnq2 tiles the group. Further, using our combinatorial methods we give a simple proof for the statement that Fuglede’s conjecture holds for p2.

Keywords
Pompeiu problem, spectral set conjecture, tiling, vanishing sums of roots of unity
Mathematical Subject Classification 2010
Primary: 39B32, 43A46
Secondary: 13F20, 20K01
Milestones
Received: 10 July 2018
Revised: 2 March 2019
Accepted: 8 April 2019
Published: 15 April 2020
Authors
Gergely Kiss
Faculty of Science
Mathematics Research Unit
University of Luxembourg
Luxembourg
Luxembourg
Romanos Diogenes Malikiosis
Department of Mathematics
Aristotle University of Thessaloniki
Thessaloniki
Greece
Gábor Somlai
MTA-ELTE Geometric and Algebraic Combinatorics Research Group NKFIH 115799 and
Faculty of Science, Institute of Mathematics
Eötvös Loránd University
Budapest
Hungary
Máté Vizer
Alfréd Rényi Institute of Mathematics
Hungarian Academy of Sciences
Budapest
Hungary