  Frames of translates of f in L^2(G) are characterized in terms of the
zero-set of the so-called spectral symbol of f in the setting of a locally
compact abelian group G having a compact open subgroup H. We refer to such a G
as a number theoretic group. This characterization was first proved in 1992 by
Shidong Li and one of the authors for L^2(R^d) with the same formal statement
of the characterization. For number theoretic groups, and these include local
fields, the strategy of proof is necessarily entirely different; and it
requires a new notion of translation that reduces to the usual definition in
R^d.
