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
MTA-ELTE Geometric and Algebraic
Combinatorics Research Group NKFIH 115799 and
Faculty of Science, Institute of Mathematics
Eötvös Loránd University
Budapest
Hungary