  We consider the discrete versions of the well known Borg theorem and use
simple linear algebraic techniques to obtain new versions of the discrete Borg
type theorems. To be precise, we prove that the periodic potential of a
discrete Schrodinger operator is almost a constant if and only if the possible
spectral gaps of the operator are of small width. This result is further
extended to more general settings and the connection to the well known Ten
Martini problem is also discussed.
