  We consider discrete periodic operator on $\mathbb Z^d$ with respect to
lattices $\Gamma\subset\mathbb Z^d$ of full rank. We describe the class of
lattices $\Gamma$ for which the operator may have a spectral gap for
arbitrarily small potentials. We also show that, for a large class of lattices,
the dimensions of the level sets of spectral band functions at the band edges
do not exceed $d-2$.
