Uniform estimates for Delannoy numbers and dimension-free estimates for discrete maximal functions over cross-polytopes

Abstract

We prove a uniform upper and lower bound for Delannoy numbers. This is achieved by using the representation of Delannoy numbers as the number of lattice points in high-dimensional cross-polytopes (also known as hyper-octahedrons or ${\ell }^1$ balls) and proving a uniform (dimension-free) count for these lattice points. Using this count, we establish dimension-free estimates for discrete maximal functions over cross-polytopes. By proving a comparison principle with the continuous setting, we obtain a dimension-free estimate on all ${\ell }^p({\mathbb{Z}}^d)$ spaces for radii $R>C{d}^{3/2}.$ We also treat the full maximal function on ${\ell }^2({\mathbb{Z}}^d)$ for small radii $R\le d^{1-\epsilon }$ and the dyadic maximal function for any radii.

22 pages