Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets

TL;DR

This paper proves that convex subsets of algebraic Delone sets in the plane can be uniquely identified using a finite number of X-ray directions, extending known lattice results to nonperiodic quasicrystal models.

Contribution

It establishes the existence of a finite set of directions for unique convex subset identification in algebraic Delone sets, generalizing lattice results to nonperiodic structures.

Findings

Four directions suffice for unique identification of convex subsets.

Existence of a natural number $c_{\Lambda}$ for convex subset distinction.

Extension of lattice results to nonperiodic quasicrystal models.

Abstract

We consider algebraic Delone sets $\varLambda$ in the Euclidean plane and address the problem of distinguishing convex subsets of $\varLambda$ by X-rays in prescribed $\varLambda$-directions, i.e., directions parallel to nonzero 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 $u$. It is shown that for any algebraic Delone set $\varLambda$ there are four prescribed $\varLambda$-directions such that any two convex subsets of $\varLambda$ can be distinguished by the corresponding X-rays. We further prove the existence of a natural number $c_{\varLambda}$ such that any two convex subsets of $\varLambda$ can be distinguished by their X-rays in any set of $c_{\varLambda}$ prescribed $\varLambda$-directions. In particular, this extends a well-known result of Gardner and Gritzmann on the corresponding problem for planar lattices to nonperiodic cases that are relevant in quasicrystallography.