Diffraction theory beyond translation-bounded measures

• Applicant: Jan Mazac
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 579756950

The main objective of this proposal is to extend the known and well-established diffraction theory for translation-bounded measures. We plan to extend it in two main directions. The first one goes beyond translation-boundedness. We want to analyse the so-called crystalline measures introduced by Meyer using the theory of mild distributions and with the help of established techniques for almost periodic functions. Recall that crystalline measures are tempered distributions whose support is a locally finite set, and the same holds for the support of their Fourier transform. As a first step, we want to analyse a particular example of such a crystalline measure, which goes back to Guinand. This distribution is not translation bounded, but when convolved with a test function with compact support, one obtains a Bohr almost periodic function. This is the starting point for our next analysis, which is planned to go in several possible directions. We want to fully understand this example, as we believe that we can directly extend it to a particular subclass of crystalline measures. We also aim for a better understanding of which conditions give rise to almost periodic functions from distributions. In particular, we are interested in the following question. Given a family of test functions, what are the distributions that are almost periodic with respect to the family? The second main objective is to apply the recently developed counting diffraction theory for measures supported on sets of zero density. We want to analyse a family of fixed points of constant-length substitutions which give rise to point sets of zero density. They seem to be tightly related to Cantor set (and fat Cantor sets), and we want to make this connection explicit, profit from it and add another layer to the known theory of aperiodic order. We expect to describe these sets as cut-and-project sets and compute their counting autocorrelation explicitly. We expect to obtain new results which go beyond the standard ones for regular model sets. We also want to analyse the so-called Behrend subshifts, for which we expect a suitable description as weak model sets. This should also serve as a starting point for the diffraction analysis of higher-dimensional d-free systems.

• DFG Programme: Position