05646269

j

05646269 Huck, Christian Uniqueness in discrete tomography of Delone sets with long-range order. Discrete Comput. Geom. 42, No. 4, 740-758 (2009). 2009 Springer-Verlag, New York, NY EN discrete tomography discrete parallel $X$-ray $U$-polygon algebraic Delone set $p$-adic valuation cyclotomic model set

doi:10.1007/s00454-009-9213-z

Summary: We address the problem of determining finite subsets of Delone sets $\Lambda \subset \Bbb R^{d}$ with long-range order by $X$-rays in prescribed $\Lambda $-directions, i.e., directions parallel to nonzero 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 $\Lambda \subset \Bbb R^{2}$ and derive a sufficient condition for the determination of the convex subsets of these sets by $X$-rays in four prescribed $\Lambda $-directions.