Discrete Analogues in Harmonic Analysis: Spherical Averages

In this paper we prove an analogue in the discrete setting of Z d,ofthe spherical maximal theorem for 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 ≥ 5, and moreover the range of p for the L p estimates differs from its analogue in R d..