  In this paper, we extend a classical vanishing result of Burnside from the
character tables of finite groups to the character tables of commutative fusion
rings, or more generally to a certain class of abelian normalizable
hypergroups. We also treat the dual vanishing result. We show that any
nilpotent fusion categories satisfy both Burnside's property and its dual.
Using Drinfeld's map, we obtain that the Grothendieck ring of any
weakly-integral modular fusion category satisfies both properties. As
applications, we prove new identities that hold in the Grothendieck ring of any
weakly-integral fusion category satisfying the dual-Burnside's property, thus
providing new categorification criteria. In particular, we improve [OY23,
Theorem 4.5] as follows: A weakly integral modular fusion category of FPdim md
with d square-free coprime with m and FPdim(X)^2 for every simple object X, has
a pointed modular fusion subcategory of FPdim d. We also prove some new results
on the perfect fusion categories.
