  We study the diagonals of two-dimensional tilings generated by direct product
substitutions. The properties of these diagonals are primarily determined by
the eigenvalues of the substitution matrix, but also the order of the letters
in the substitution plays a role. We show that the diagonals may fail to be
uniformly recurrent, and that the frequencies of letters on the diagonal may
not exist. We also highlight the connection with the density of coincidences
and overlap distributions.
