  We initiate the study of the $\ell^p(\mathbb{Z}^d)$-boundedness of the
arithmetic spherical maximal function over sparse sequences. We state a
folklore conjecture for lacunary sequences, a key example of Zienkiewicz and
prove new bounds for a family of sparse sequences that achieves the endpoint of
the Magyar--Stein--Wainger theorem for the full discrete spherical maximal
function. Perhaps our most interesting result is the boundedness of a discrete
spherical maximal function in $\mathbb{Z}^4$ over an infinite, albeit sparse,
set of radii. Our methods include the Kloosterman refinement for the Fourier
transform of the spherical measure (introduced by Magyar) and Weil bounds for
Kloosterman sums which are utilized by a new further decomposition of spherical
measure.
