Report
Teaser, summary, work performed and final results
Periodic Reporting for period 2 - HARMONIC (Studies in Harmonic Analysis and Discrete Geometry: Tilings, Spectra and Quasicrystals)
Teaser
The project is concerned with several themes which lie in the crossroads of Harmonic Analysis and Discrete Geometry. Harmonic Analysis is fundamental in all areas of science and engineering, and has vast applications in most branches of mathematics. Discrete Geometry deals...
Summary
The project is concerned with several themes which lie in the crossroads of Harmonic Analysis and Discrete Geometry. Harmonic Analysis is fundamental in all areas of science and engineering, and has vast applications in most branches of mathematics. Discrete Geometry deals with some of the most natural and beautiful problems in mathematics, which often turn out to be also very deep and difficult in spite of their apparent simplicity. The project deals with some fundamental problems which involve an interplay between these two important disciplines. One theme of the project is concerned with tilings of the Euclidean space by translations, and the interaction of this subject with questions in orthogonal harmonic analysis, such as the famous conjecture due to Fuglede (1974) concerning the characterization of domains which admit orthogonal Fourier bases in terms of their possibility to tile the space by translations, and in relation with the theory of multiple tiling by translates of a domain or a function. Another theme of the research lies in the mathematical theory of quasicrystals. This area has received a lot of attention since the experimental discovery in the 1980\'s of the physical quasicrystals, namely, of non-periodic atomic structures with diffraction patterns consisting of spots. In the present project, we investigate the geometry and structure of these rigid point configurations, and analyze some basic problems which are still open in this field.
Work performed
We have succeeded to advance significantly on several themes central to the project.
Fourier quasicrystals: In a joint paper with Alexander Olevskii we analyse the structure of quasicrystals under non-symmetric discreteness assumptions on the support and the spectrum. The main situation considered involves quasicrystals with uniformly discrete support and locally finite spectrum. We obtained several results which show that, under various assumptions, the quasicrystal must have a periodic structure.
Fuglede’s spectral set conjecture: In a joint paper with Rachel Greenfeld we obtained a proof of the “spectral implies tiling†part of Fuglede’s spectral set conjecture for convex polytopes in dimension three. This result confirms that the conjecture is true for three-dimensional convex polytopes. In a subsequent joint paper with Rachel Greenfeld we obtained several other results which in particular support the conjecture in dimensions 4 and higher. Finally in a recent preprint joint with Máté Matolcsi we prove the Fuglede conjecture for convex domains in all dimensions. In a joint paper with Bochen Liu we prove that any non-convex spectral polytope must be equidecomposable by translations to a cube, and hence it “nearly†tiles by translations.
Multi-tiling: In a joint paper with Bochen Liu we obtained a complete characterization of the polytopes in d-dimensional Euclidean space which multi-tile the space by translations along a given lattice. We also obtained a criterion for two polytopes to be equidecomposable by lattice translates. The characterizations are stated in terms of Hadwiger functionals.
Fourier frames: We have disproved a conjecture that was believed for some time concerning singular continuous measures which admit a frame of exponential functions. The conjecture stated that such a measure cannot have components of different dimensions. We established that this is not the case by proving a result which allows one to construct many examples of “mixed type†measures that have a Fourier frame.
Final results
In the second half of the project we plan to address the problem of classifying all the possible spectra of a convex polytope in d-dimensional Euclidean space. It has been known that a discrete set constitutes a spectrum for the unit cube if and only it can serve as a translation set for tiling by the cube. This result provides a complete description of all the possible spectra of the unit cube by a geometric condition. We are interested to find an extension of this result to all the spectral convex polytopes. In particular, we expect that the methods that will be developed in the research will enable us to classify the convex polytopes which admit a unique (up to translation) spectrum.
Another one of our forthcoming goals is to carry out a detailed analysis of the phenomenon discovered by Mikhail Kolountzakis and the PI, of the existence of non-periodic tilings of the real line by translates of a function with an unbounded support. We intend to continue this investigation in several directions. One of them is concerned with the question whether our result may be strengthened by taking the function to be an indicator function.
We also expect to obtain further progress in the framework of so-called Fourier quasicrystals. For example, it remains unknown whether there exists a non-periodic measure on the real line whose support is a uniformly discrete set, while the spectrum is a discrete, closed set. We expect that by elaboration and development of our methods and techniques we can shed new light on these and related directions.
Website & more info
More info: http://u.math.biu.ac.il/.