The inverse problem of pure point diffraction -- examples and open questions

TL;DR

This paper explores the inverse problem of pure point diffraction, examining which objects can produce specific diffraction patterns and presenting examples of diverse objects with identical diffraction, including non-measure distributions.

Contribution

It introduces new examples of objects with the same diffraction pattern, including non-measure distributions, and discusses the complexity of the inverse diffraction problem.

Findings

Multiple objects can produce identical pure point diffraction patterns.

There are uncountably many objects in the diffraction solution class for certain measures.

Some objects with the same diffraction are not measures, including tempered distributions.

Abstract

This paper considers some open questions related to the inverse problem of pure point diffraction, in particular, what types of objects may diffract, and which of these may exhibit the same diffraction. Some diverse objects with the same simple lattice diffraction are constructed, including a tempered distribution that is not a measure, and it is shown that there are uncountably many such objects in the diffraction solution class of any pure point diffraction measure with an infinite supporting set.