Compact open spectral sets in QpQp

In this article, we prove that a compact open set in the field QpQp of p  -adic numbers is a spectral set if and only if it tiles QpQp by translation, and also if and only if it is p  -homogeneous which is easy to check. We also characterize spectral sets in Z/pnZZ/pnZ (p≥2p≥2 prime, n≥1n≥1 integer) by tiling property and also by homogeneity. Moreover, we construct a class of singular spectral measures in QpQp, some of which are self-similar measures.