Fuglede's conjecture holds in ℤ_p×ℤ_pⁿ

2021/09/17 by Zhang, Tao
#05B25 #11P99 #42B05 #43A40 #52C20 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2109.08400

Abstract

Fuglede's conjecture states that for a subset Ω of a locally compact abelian group G with positive and finite Haar measure, there exists a subset of the dual group of G which is an orthogonal basis of L²(Ω) if and only if it tiles the group by translation. In this paper, we prove a divisibility property for a set in ℤ_p×ℤ_pⁿ. Then using the divisibility property and equi-distributed property, we prove that Fuglede's conjecture holds in the group ℤ_p×ℤ_pⁿ.

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