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.
