Mario Kummer: Fourier quasicrystals and almost periodic sets of toral type

Abstract: A function \(f\colon\mathbb{R}^n\to\mathbb{R}\) is called (Bohr) almost periodic if for every \(\epsilon>0\) the set of
\(\epsilon\)-almost periods, i.e.~\(T\in\mathbb{R}^n\) such that \(|f(x+T)-f(x)|<\epsilon\) for all \(x\in\mathbb{R}^n\), is relatively dense in \(\mathbb{R}^n\). We will recall the classical theory of how almost periodic functions can be understood in terms of continuous functions on compact abelian groups. Some of these results transfer to almost periodic discrete sets \(\Lambda\subset\mathbb{R}^n\), which can be defined in a similar vein. However, a complete geometric characterization is only known when the group is a torus \(\mathbb{T}^n\), due to Wayne Lawton. Such sets are said to be of toral type. We will report on joint work with Lior Alon where we prove that every uniformly discrete Fourier quasicrystal \(\Lambda\subset\mathbb{R}^n\) is almost periodic of toral type.