  Let $A = (a_{j,k})_{j,k=-\infty}^\infty$ be a bounded linear operator on
$l^2(\mathbb{Z})$ whose diagonals $D_n(A) = (a_{j,j-n})_{j=-\infty}^\infty\in
l^\infty(\mathbb{Z})$ are almost periodic sequences. For certain classes of
such operators and under certain conditions, we are going to determine the
asymptotics of the determinants $\det A_{n_1,n_2}$ of the finite sections of
the operator $A$ as their size $n_2 - n_1$ tends to infinity. Examples of such
operators include block Toeplitz operators and the almost Mathieu operator.
