Quasiperiodic tilings are deterministic but non-periodic structures. A Hamiltonian defined on such tilings displays a fractal (Cantor-like) eigenvalue spectrum characterized with an infinite number of gaps. Each of these gaps is labeled by a finite set of integers of topological nature. In this work, we present some results about the relation between these topological numbers and the diffraction pattern.

A common way to create a quasiperiodic tiling is with Cut and Project scheme. We define an atomic density by inserting an atom on each boundary between tiles. The corresponding structure factor, given by the absolute value of the Fourier transform of the atomic density, displays a pure-point spectrum with Bragg peaks only. Their locations in the k-space can be enumerated similarly to the gap labeling. We show that the phase of the atomic density Fourier transform is described using the previously defined spectral topological numbers. Recent experimental endeavors confirm these results, and allow to describe these topological numbers as winding numbers.