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

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

2007-11-28

arxiv.org/abs/0711.4525v2

Mathematics

Metric Geometry

15 pages, 2 figures; condensed and revised version

Scientific paper

We address the problem of determining finite subsets of Delone sets $\varLambda\subset\R^d$ with long-range order by $X$-rays in prescribed $\varLambda$-directions, i.e., directions parallel to non-zero interpoint vectors of $\varLambda$. 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 $\varLambda\subset\R^2$ and derive a sufficient condition for the determination of the convex subsets of these sets by $X$-rays in four prescribed $\varLambda$-directions.

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