  In this paper we approximate high-dimensional functions $f\colon\mathbb
T^d\to\mathbb C$ by sparse trigonometric polynomials based on function
evaluations. Recently it was shown that a dimension-incremental sparse Fourier
transform (SFT) approach does not require the signal to be exactly sparse and
is applicable in this setting. We combine this approach with subsampling
techniques for rank-1 lattices. This way our approach benefits from the
underlying structure in the sampling points making fast Fourier algorithms
applicable whilst achieving the good sampling complexity of random points
(logarithmic oversampling). In our analysis we show detection guarantees of the
frequencies corresponding to the Fourier coefficients of largest magnitude. In
numerical experiments we make a comparison to full rank-1 lattices and
uniformly random points to confirm our findings.
