Uniform estimates for Delannoy numbers and dimension-free estimates for discrete maximal functions over cross-polytopes
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| Format: | Preprint |
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2026
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| _version_ | 1866914482700156928 |
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| author | Kosz, Dariusz Niksiński, Jakub Wróbel, Błażej |
| author_facet | Kosz, Dariusz Niksiński, Jakub Wróbel, Błażej |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_15844 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform estimates for Delannoy numbers and dimension-free estimates for discrete maximal functions over cross-polytopes Kosz, Dariusz Niksiński, Jakub Wróbel, Błażej Number Theory Classical Analysis and ODEs Combinatorics 05A16, 11B75, 42B25 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. |
| title | Uniform estimates for Delannoy numbers and dimension-free estimates for discrete maximal functions over cross-polytopes |
| topic | Number Theory Classical Analysis and ODEs Combinatorics 05A16, 11B75, 42B25 |
| url | https://arxiv.org/abs/2604.15844 |