  In this paper we prove an analogue in the discrete setting of \Bbb Z^d, of
the spherical maximal theorem for \Bbb R^d. The methods used are two-fold: the
application of certain "sampling" techniques, and ideas arising in the study of
the number of representations of an integer as a sum of d squares in
particular, the "circle method". The results we obtained are by necessity
limited to d \ge 5, and moreover the range of p for the L^p estimates differs
from its analogue in \Bbb R^d.
