#### 1.6 Pure point spectra in diffraction from quasicrystals.

Sharp diffraction peaks in scattering are found not only in crystals but also in
quasicrystals. The reason for this is given in Harald Bohr’s theory of quasiperiodic
functions.

Bohr [2], [3] from his concept of restrictions to irrational sections E_{∥}^{m} < E^{n},m < n
eq. 8 of periodic functions derived the Fourier properties of quasiperiodic
functions. This requires the notion of a Z-module.

Under projection, the parallel projected lattice points of Λ fall on a Z-module.
The points of this Z-module are the integral linear combinations of the
projections of the lattice basis vectors from E^{n} to E_{
∥}^{m}. The number of lattice
basis vectors from Λ E^{n} is n and so is the number of projected basis
vectors of the Z-module in E^{m},m < n. These n projections cannot be
linear independent on E^{m} w.r.t. real coefficients. They are nevertheless
linear independent w.r.t. integral linear combinations. As a consequence,
the set of these module points is discrete and countable but dense on
E^{m}.

The periodic functions on E^{n}, as we have seen in section 2.3, have their Fourier
coefficients on a reciprocal lattice Λ^{R} in k-space E^{n}. By Fourier-analysing the
restriction of the periodic function it can be shown that the Fourier spectrum of a
quasiperiodic function is discrete. Taking into account the projection,
the Fourier coefficients of a quasiperiodic function can be assigned to a
Z-module on a k-space E_{∥}^{m}, with module points the parallel projections of the
reciprocal lattice Λ^{R} from the k-space E^{n} to a k-space E_{
∥}^{m}. Again the
module points are dense. Fortunately, the observed Fourier coefficients vary
strongly in intensity and therefore still allow for their assignment to discrete
module points. It is now standard in quasicrystallography to assign the
diffraction peaks to reciprocal lattice points in k-space E^{n} or to their parallel
projections.