Uniform bounds on Delannoy numbers from lattice-point counts in cross-polytopes produce dimension-free estimates for discrete maximal functions over these polytopes for radii larger than order d to the 3/2.
Carbery,An almost-orthogonality principle with applications to maximal functions asso- ciated to convex bodies, Bull
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Uniform estimates for Delannoy numbers and dimension-free estimates for discrete maximal functions over cross-polytopes
Uniform bounds on Delannoy numbers from lattice-point counts in cross-polytopes produce dimension-free estimates for discrete maximal functions over these polytopes for radii larger than order d to the 3/2.