New bounds for discrete lacunary spherical averages

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.