We show that the discrete lacunary spherical maximal function is bounded on
lp(Zd) for all p>dβ1d+1β. 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.Comment: Errors corrected. Proof is restructured, but main result remains the
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