  We give necessary and sufficient conditions for a subfamily of regularly
spaced translates of a function to form a frame (resp. a Riesz basis) for its
span. One consequence is that ifthetranslates are taken only from a subset of
the natural numbers, then this family is a frame if and only if it is a Riesz
basis. We also consider arbitrary sequences of translates and show that for
sparse sets, having an upper frame bound is equivalent to the family being a
frame sequence. Finally, we use the fractional Hausdorff dimension to identify
classes of exact frame sequences.
