We prove sparse bounds for the spherical maximal operator of Magyar, Stein and
Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The
new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient,
and has not been used in the proof of sparse bounds before. The Hardy–Littlewood
circle method is used to decompose the multiplier into major and minor arc
components. The efficiency arises as one only needs a single estimate on each element
of the decomposition.
Keywords
sparse bounds, spherical averages, discrete, spherical
maximal function