In this paper, we prove that given a cut-and-project scheme $(G, H, \mathcal {L})$ and a compact window $W \subseteq H$, the natural projection gives a bijection between the Fourier transformable measures on $G \times H$ supported inside the strip ${\mathcal L} \cap (G \times W)$ and the Fourier transformable measures on G supported inside ${\LARGE \curlywedge }(W)$. We provide a closed formula relating the Fourier transform of the original measure and the Fourier transform of the projection. We show that this formula can be used to re-derive some known results about Fourier analysis of measures with weak Meyer set support.

We give a characterization of inter-model sets with Euclidean internal space. This characterization is similar to previous results for general inter-model sets obtained independently by Baake, Lenz and Moody, and Aujogue. The new ingredients are two additional conditions. The first condition is on the rank of the abelian group generated by the set of internal differences. The second condition is on a flow on a torus defined via the address map introduced by Lagarias. This flow plays the role of the maximal equicontinuous factor in the previous characterizations.

In this paper we study the existence of higher dimensional arithmetic progressions in Meyer sets. We show that the case when the ratios are linearly dependent over ${\mathbb Z}$ is trivial and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set $\Lambda $ and a fully Euclidean model set with the property that finitely many translates of cover $\Lambda $ , we prove that we can find higher dimensional arithmetic progressions of arbitrary length with k linearly independent ratios in $\Lambda $ if and only if k is at most the rank of the ${\mathbb Z}$ -module generated by . We use this result to characterize the Meyer sets that are subsets of fully Euclidean model sets.