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 \(d-2\) .
