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

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Abstract

We address the problem of determining finite subsets of Delone sets Λ⊂ℝd with long-range order by X-rays in prescribed Λ-directions, i.e., directions parallel to nonzero interpoint vectors of Λ. 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 Λ⊂ℝ2 and derive a sufficient condition for the determination of the convex subsets of these sets by X-rays in four prescribed Λ-directions.

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Author information

Correspondence to Christian Huck.

Additional information

The author was supported by the German Research Council (Deutsche Forschungsgemeinschaft), within the CRC 701, and by EPSRC via Grant EP/D058465/1.

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Huck, C. Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order. Discrete Comput Geom 42, 740 (2009). https://doi.org/10.1007/s00454-009-9213-z

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Keywords

  • Discrete tomography
  • Discrete parallel X-ray
  • U-polygon
  • Algebraic Delone set
  • p-adic valuation
  • Cyclotomic model set