We study discretized maximal operators associated to averaging over (neighborhoods of) squares in the plane and, more generally, k-skeletons in Rn. Although these operators are known not to be bounded on any Lp, we obtain nearly sharp Lp bounds for every small discretization scale. These results are motivated by, and partially extend, recent results of Keleti, Nagy and Shmerkin, and of Thornton, on sets that contain a scaled k-sekeleton of the unit cube with center in every point of Rn.
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