  We show that the discrete lacunary spherical maximal function is bounded on
$l^p(\mathbb{Z}^d)$ for all $p >\frac{d+1}{d-1}$. Our range is new in dimension
4, where it appears that little was previously known for general lacunary
radii. Our technique follows that of Kesler-Lacey-Mena, using the Kloosterman
refinement to improve the estimates in several places, which leads to an
overall improvement in dimension 4.
