Bases of exponentials and tilings

Analysis Seminar
Wednesday, September 4, 2019 - 1:55pm for 1 hour (actually 50 minutes)
Skiles 005
Mihalis Kolountzakis – University of Crete – kolount@gmail.com
Shahaf Nitzan

Mathematicians have long been trying to understand which domains admit an orthogonal (or, sometimes, not) basis of exponentials of the form , for some set of frequencies (this is the spectrum of the domain). It is well known that we can do so for the cube, for instance (just take ), but can we find such a basis for the ball? The answer is no, if we demand orthogonality, but this problem is still open when, instead of orthogonality, we demand just a Riesz basis of exponentials.

This question has a lot to do with tiling by translation (i.e., with filling up space with no overlaps by translating around an object). Fuglede originally conjectured that an orthogonal exponential basis exists if and only if the domain can tile space by translation. This has been disproved in its full generality but when one adds side conditions, such as, for instance, a lattice set of frequencies, or the space being a group of a specific type, or many other natural conditions, the answer is often unknown, and sometimes known to be positive or known to be negative. A major recent  development is the proof (2019) by Lev and Matolcsi of the truth of the Fuglede conjecture for convex bodies in all dimensions.
This is a broad area of research, branching out by varying the side conditions on the domain or the group in which the domain lives, or by relaxing the orthogonality condition or even allowing time-frequency translates of a given function to serve as basis elements (Gabor, or Weyl-Heisenberg, bases). When working with both exponential bases and tiling problems the crucial object of study turns out to be the zero set of the Fourier Transform of the indicator function of the domain we care about. In particular we want to know how large structured sets this zero set contains, for instance how large difference sets it contains or what kind of tempered distributions it can support.
In this talk I will try to show how these objects are tied together, what has been done recently, and indicate specific open problems.