In this talk we prove a discrete version of the Bethe-Sommerfeld conjecture. 

Namely, we show that the spectra of multi-dimensional discrete periodic Schr\"odinger operators on ℤd​ lattice with sufficiently small potentials contain at most two intervals. 

Moreover, the spectrum is a single interval, provided one of the periods is odd, and can have a gap whenever all periods are even. This is based on a joint work with Lana.