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

TL;DR

This paper establishes uniform bounds for Delannoy numbers via lattice point counts in high-dimensional cross-polytopes and derives dimension-free estimates for discrete maximal functions over these shapes.

Contribution

It introduces a uniform lattice point count for cross-polytopes and applies this to obtain dimension-free estimates for discrete maximal functions in various settings.

Findings

Uniform upper and lower bounds for Delannoy numbers.

Dimension-free estimates for discrete maximal functions over cross-polytopes.

Results applicable to all 1(2^d) spaces for large radii.

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 $R\le d^{1-\varepsilon}$ and the dyadic maximal function for any radii.