Abstract
In this paper we prove an analogue in the discrete setting of $\mathbb{Z}^d$, of the spherical maximal theorem for $\mathbb{R}^d$. 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 necessary limited to $d\ge 5$, and moreover the range of $p$ for the $L^p$ estimates differs from its analogue in $\mathbb{R}^d$.
