  Given a subset of the integers of zero density, we define the weaker notion
of fractional density of such a set. It is shown how this notion corresponds to
that of the Hausdorff dimension of a compact subset of the reals. We then show
that a version of a theorem of {\L}aba and Pramanik on 3-term arithmetic
progressions in subsets of the unit interval also holds for subsets of the
integers with fractional density and satisfying certain Fourier-decay
conditions.
