Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order

We address the problem of determining finite subsets of Delone sets $\Lambda \subset {\mathbb{R}}^d$ with long-range order by $X$-rays in prescribed $\Lambda$-directions, i.e., directions parallel to non-zero interpoint vectors of $\Lambda$. Here, an $X$-ray in direction $u$ of a finite set gives the number of points in the set on each line parallel to $u$. For our main result, we introduce the notion of algebraic Delone sets $\Lambda \subset {\mathbb{R}}^2$ and derive a sufficient condition for the determination of the convex subsets of these sets by $X$-rays in four prescribed $\Lambda$-directions.