  Arithmetic progressions of length $3$ may be found in compact subsets of the
reals that satisfy certain Fourier -- as well as Hausdorff -- dimensional
requirements. It has been shown that a very similar result holds 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 $\mathbb{Z}^n$ satisfying
appropriate density and Fourier-decay conditions.
