Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups

We prove the existence of sampling sets and interpolation sets near the critical density, in Paley Wiener spaces of a locally compact abelian (LCA) group G . This solves a problem left by Gr\"ochenig, Kutyniok, and Seip in the article: `Landau's density conditions for LCA groups ' (J. of Funct. Anal. 255 (2008) 1831-1850). To achieve this result, we prove the existence of universal Riesz bases of characters for L2(Omega), provided that the relatively compact subset Omega of the dual group of G satisfies a multi-tiling condition. This last result generalizes Fuglede's theorem, and extends to LCA groups setting recent constructions of Riesz bases of exponentials in bounded sets of Rd.