  We obtain positive and negative results concerning lacunary discrete maximal
operators defined by dilations of sufficiently nonsingular hypersurfaces
arising from Diophantine equations in many variables. Our negative results show
that this problem differs substantially from that of lacunary discrete maximal
operators defined along a nonsingular hypersurface. Our positive results are
improvements over bounds for the corresponding full maximal functions which
were initially studied by Magyar.
  In order to obtain positive results, we use an interpolation technique of the
second author to reduce problem to a maximal function of main terms. The main
terms take the shape of those introduced in work of the first author, which is
a more localized version of the main terms that appear in work of Magyar. The
main ingredient of this paper is a new bound on the main terms near $\ell^1$.
For our negative results we generalize an argument of Zienkiewicz.
