  We demonstrate $k+1$-term arithmetic progressions in certain subsets of the
real line whose "higher-order Fourier dimension" is sufficiently close to 1.
This Fourier dimension, introduced in previous work, is a higher-order (in the
sense of Additive Combinatorics and uniformity norms) extension of the Fourier
dimension of Geometric Measure Theory, and can be understood as asking that the
uniformity norm of a measure, restricted to a given scale, decay as the scale
increases. We further obtain quantitative information about the size and $L^p$
regularity of the set of common distances of the artihmetic progressions
contained in the subsets of $\mathbb{R}$ under consideration.
