Periodic Reporting for period 4 - HARMONIC (Studies in Harmonic Analysis and Discrete Geometry: Tilings, Spectra and Quasicrystals)
Période du rapport: 2021-06-01 au 2022-11-30
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.
Fourier quasicrystals: The PI with Alexander Olevskii analysed 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. The results obtained show that, under various assumptions, the quasicrystal must have a periodic structure.
Multi-tiling: The PI with Bochen Liu obtained a complete characterization of the polytopes in d-dimensional Euclidean space which multi-tile the space by translations along a given lattice. They also obtained a criterion for two polytopes to be equidecomposable by lattice translates. The characterizations are stated in terms of Hadwiger functionals.
Fourier frames: The PI 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. The PI 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.
Fuglede's spectral set conjecture: The PI with Mate Matolcsi obtained a proof of the "spectral implies tiling" part of Fuglede's spectral set conjecture for convex domains in all dimensions. The result fully settles the Fuglede conjecture for convex bodies affirmatively in all dimensions. This solved one of the central problems of the ERC project and answers a long-standing question in the area.
Spectrality and equidecomposability by translations: The PI with Bochen Liu extended significantly a result due to Kolountzakis and Papadimitrakis (2002) which gives a necessary condition for the spectrality of non-convex polytopes. The result was extended from faces of codimension one to faces of all dimensions. As a consequence it was established that any non-convex spectral polytope must be equidecomposable by translations to a cube.
Riesz bases of exponentials: The PI with Alberto Debernardi obtained a proof that for any convex polytope in d-dimensional space which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions.
Crystalline temperate distributions: The PI with Gilad Reti analysed the structure of crystalline temperate distributions on the real line with uniformly discrete support and spectrum. They proved that any temperate distribution of this type can be obtained from the classical Poisson summation formula by a finite number of basic operations (shifts, modulations, differentiations, multiplication by polynomials, and taking linear combinations).
Tiling by translates of a function: The PI with Mihalis Kolountzakis addressed some open problems concerned with tilings of the real line by translates of a function. The phenomena investigated include tilings of bounded and of unbounded density, periodic and non-periodic tilings, and tilings at level zero.
Gabor orthonormal bases: The PI with Alberto Debernardi showed that if the Gabor system G(g,T,S) is an orthonormal basis in L2(R) and if the window function g is compactly supported, then both the time shift set T and the frequency shift set S must be periodic. To prove this, they established a necessary functional tiling type condition for Gabor orthonormal bases which is also of independent interest.
Simultaneous tiling by translates of a function: The PI with Mark Mordechai Etkind obtained sharp results on the least possible size of the support of a function that tiles simultaneously by two arithmetic progressions.
Multi-tiling: The PI with Bochen Liu obtained a complete characterization of the polytopes in d-dimensional Euclidean space which multi-tile the space by translations along a given lattice. They also obtained a criterion for two polytopes to be equidecomposable by lattice translates. The characterizations are stated in terms of Hadwiger functionals.
Fourier frames: The PI 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. The PI 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.
Fuglede's spectral set conjecture: The PI with Mate Matolcsi obtained a proof of the "spectral implies tiling" part of Fuglede's spectral set conjecture for convex domains in all dimensions. The result fully settles the Fuglede conjecture for convex bodies affirmatively in all dimensions. This solved one of the central problems of the ERC project and answers a long-standing question in the area.
Spectrality and equidecomposability by translations: The PI with Bochen Liu extended significantly a result due to Kolountzakis and Papadimitrakis (2002) which gives a necessary condition for the spectrality of non-convex polytopes. The result was extended from faces of codimension one to faces of all dimensions. As a consequence it was established that any non-convex spectral polytope must be equidecomposable by translations to a cube.
Riesz bases of exponentials: The PI with Alberto Debernardi obtained a proof that for any convex polytope in d-dimensional space which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions.
Crystalline temperate distributions: The PI with Gilad Reti analysed the structure of crystalline temperate distributions on the real line with uniformly discrete support and spectrum. They proved that any temperate distribution of this type can be obtained from the classical Poisson summation formula by a finite number of basic operations (shifts, modulations, differentiations, multiplication by polynomials, and taking linear combinations).
Tiling by translates of a function: The PI with Mihalis Kolountzakis addressed some open problems concerned with tilings of the real line by translates of a function. The phenomena investigated include tilings of bounded and of unbounded density, periodic and non-periodic tilings, and tilings at level zero.
Gabor orthonormal bases: The PI with Alberto Debernardi showed that if the Gabor system G(g,T,S) is an orthonormal basis in L2(R) and if the window function g is compactly supported, then both the time shift set T and the frequency shift set S must be periodic. To prove this, they established a necessary functional tiling type condition for Gabor orthonormal bases which is also of independent interest.
Simultaneous tiling by translates of a function: The PI with Mark Mordechai Etkind obtained sharp results on the least possible size of the support of a function that tiles simultaneously by two arithmetic progressions.
The PI with Mate Matolcsi introduced a novel methodology in order to prove the Fuglede conjecture for convex domains in all dimensions. The method involves a construction from crystallographic diffraction theory, that allowed to establish a geometric "weak tiling" condition necessary for the spectrality of an arbitrary (not necessarily convex) domain.
The PI with Rachel Greenfeld introduced new methodology that allowed to prove that certain polytopes admit a unique spectrum, and to make progress on the spectrality problem for product domains. The methods use discrete combinatorial considerations combined with Fourier analytic ones.
The PI with Bochen Liu developed significantly a method introduced by Kolountzakis and Papadimitrakis (2002) that allowed to extend a necessary condition for the spectrality of a non-convex polytope from faces of codimension one to faces of all dimensions. The approach involves a delicate analysis of the asymptotics of the Fourier transform of the indicator function of a (not necessarily convex) polytope.
The PI gave a novel method that allowed to construct many examples of "mixed type" measures that admit a Fourier frame. This was believed for some time to be not possible. The method involves a new result which asserts that if two continuous singular measures admit (each one separately) a Fourier frame and if the measures are supported in two distinct orthogonal subspaces, then their sum also admits a Fourier frame.
The PI with Mark Mordechai Etkind developed a new approach that gives sharp results on the least possible measure of the support of a function that tiles simultaneously with two arithmetic progressions. The approach is graph-theoretic and is based on the observation that any simultaneously tiling function induces a weighted graph, whose vertices and edges can also be endowed with a measure space structure. The main method is an iterative leaves removal process that is applied to this graph.
The proofs of these results involve a combination of methods from harmonic analysis, discrete geometry, mathematical crystallography and combinatorics. The methods are likely to find further applications in these and related areas.
The PI with Rachel Greenfeld introduced new methodology that allowed to prove that certain polytopes admit a unique spectrum, and to make progress on the spectrality problem for product domains. The methods use discrete combinatorial considerations combined with Fourier analytic ones.
The PI with Bochen Liu developed significantly a method introduced by Kolountzakis and Papadimitrakis (2002) that allowed to extend a necessary condition for the spectrality of a non-convex polytope from faces of codimension one to faces of all dimensions. The approach involves a delicate analysis of the asymptotics of the Fourier transform of the indicator function of a (not necessarily convex) polytope.
The PI gave a novel method that allowed to construct many examples of "mixed type" measures that admit a Fourier frame. This was believed for some time to be not possible. The method involves a new result which asserts that if two continuous singular measures admit (each one separately) a Fourier frame and if the measures are supported in two distinct orthogonal subspaces, then their sum also admits a Fourier frame.
The PI with Mark Mordechai Etkind developed a new approach that gives sharp results on the least possible measure of the support of a function that tiles simultaneously with two arithmetic progressions. The approach is graph-theoretic and is based on the observation that any simultaneously tiling function induces a weighted graph, whose vertices and edges can also be endowed with a measure space structure. The main method is an iterative leaves removal process that is applied to this graph.
The proofs of these results involve a combination of methods from harmonic analysis, discrete geometry, mathematical crystallography and combinatorics. The methods are likely to find further applications in these and related areas.