On the discrete Fuglede and Pompeiu problems

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

doi:10.2140/apde.2020.13.765

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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 $ℤ_{p^{n} q^{2}}$ , where $p$ and $q$ are different primes. In particular, we show that every spectral subset of $ℤ_{p^{n} q^{2}}$ tiles the group. Further, using our combinatorial methods we give a simple proof for the statement that Fuglede’s conjecture holds for $ℤ_{p}^{2}$ .

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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

Vol. 13, No. 3, 2020