Lacunary Discrete Spherical Maximal Functions

We prove new $ell ^{p} (mathbb Z ^{d})$ bounds for discrete spherical averages in dimensions $ d geq 5$. We focus on the case of lacunary radii, first for general lacunary radii, and then for certain kinds of highly composite choices of radii. In particular, if $ A _{lambda } f $ is the spherical average of $ f$ over the discrete sphere of radius $ lambda $, we have begin{equation*} bigllVert sup _{k} lvert A _{lambda _k} f rvert bigrrVert _{ell ^{p} (mathbb Z ^{d})} lesssim lVert frVert _{ell ^{p} (mathbb Z ^{d})}, qquad tfrac{d-2} {d-3} < p leq tfrac{d} {d-2}, dgeq 5, end{equation*} for any lacunary sets of integers $ {lambda _k ^2 }$. We follow a style of argument from our prior paper, addressing the full supremum. The relevant maximal operator is decomposed into several parts; each part requires only one endpoint estimate.