Arithmetic Progressions in Salem-Type Subsets of the Integers

Given a subset of the integers of zero density, we define the weaker notion of the fractional density of such a set. We show that a version of a theorem of Łaba and Pramanik on 3-term arithmetic progressions in subsets of the unit interval also holds for subsets of the integers with fractional density whose characteristic functions have Fourier coefficients that decay sufficiently rapidly.