Arithmetic progressions of length three may be found in compact subsets of the reals that satisfy certain Fourier- as well as Hausdorff-dimensional requirements. Similar results hold in the integers under analogous conditions, with Fourier dimension being replaced by the decay of a discrete Fourier transform. In this paper we make this correspondence more precise, using a well-known construction by Salem. Specifically, we show that a subset of the integers can be mapped to a compact subset of the continuum in a way which preserves certain dimensional properties as well as arithmetic progressions of arbitrary length. The higher-dimensional version of this construction is then used to show that certain parallelogram configurations must exist in sparse subsets of Zⁿ satisfying appropriate density and Fourier-decay conditions.