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We discuss the relation of tiling, weak tiling and spectral sets in finiteabelian groups. In particular, in elementary $p$-groups $(\mathbb{Z}_p)^d$, weintroduce an averaging procedure that leads to a natural object of study: a4-tuple of functions which can be regarded as a common generalization of tilesand spectral sets. We characterize such 4-tuples for $d=1, 2$, and prove somepartial results for $d=3$.

Tiling and weak tiling in $(\mathbb{Z}_p)^d$