Compact open spectral sets in Q(p)

International audienceIn this article, we prove that a compact open set in the field Q(p) of p-adic numbers is a spectral set if and only if it tiles Q(p) by translation, and also if and only if it is p-homogeneous which is easy to check. We also characterize spectral sets in Z/p(n)Z (p >= 2 prime, n >= 1 integer) by tiling property and also by homogeneity. Moreover, we construct a class of singular spectral measures in Q(p), some of which are self similar measures. (C) 2016 Elsevier Inc. All rights reserved