  Limit periodic point sets are aperiodic structures with pure point
diffraction supported on a countably, but not finitely generated Fourier module
that is based on a lattice and certain integer multiples of it. Examples are
cut and project sets with p-adic internal spaces. We illustrate this by
explicit results for the diffraction measures of two examples with 2-adic
internal spaces. The first and well-known example is the period doubling
sequence in one dimension, which is based on the period doubling substitution
rule. The second example is a weighted planar point set that is derived from
the classic chair tiling in the plane. It can be described as a fixed point of
a block substitution rule.
