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

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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 Gröchenig, 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.