  We propose a tunable electronic band gap and zero-energy modes in periodic
heterosubstrate-induced graphene superlattices. Interestingly, there is an
approximate linear relation between the band gap and the proportion of
inhomogeneous substrate (i.e., percentages of different components) in the
proposed superlattice, and the effect of structural disorder on the relation is
discussed. In inhomogeneous substrate with equal widths, zero-energy states
emerge in the form of Dirac points by using asymmetric potentials, and the
positions of Dirac points are addressed analytically. Further, the Dirac point
exists at $\mathbf{k}=\mathbf{0}$ only for specific potentials; every time it
appears, the group velocity vanishes in $k_y$ direction and the resonance
occurs. For general cases that inhomogeneous substrate with unequal widths, a
part of zero-energy states are described analytically, and differently, they
are not always Dirac points. Our prediction may be realized on the
heterosubstrate such as SiO$_2$/BN type.
