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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2504.14424 |
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| _version_ | 1866908328323448832 |
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| author | Lott, Andrew Magyar, Ákos Petridis, Giorgis Pintz, János |
| author_facet | Lott, Andrew Magyar, Ákos Petridis, Giorgis Pintz, János |
| contents | We provide a multidimensional extension of previous results on the existence of polynomial progressions in dense subsets of the primes. Let $A$ be a subset of the prime lattice - the d-fold direct product of the primes - of positive relative upper density. We show that A contains all polynomial configurations of the form $x+P_0(y)v_0,\ldots, x+P_l(y)v_l$, for some $x$ in $\mathbb{Z}^d$ and $y$ in $\mathbb{N}$, which satisfy a certain non-degeneracy condition. We also obtain quantitative bounds on the size of such polynomial configuration, if $A$ is a subset of the first $N$ positive integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial configurations in dense subsets of the prime lattice Lott, Andrew Magyar, Ákos Petridis, Giorgis Pintz, János Number Theory 11N05 We provide a multidimensional extension of previous results on the existence of polynomial progressions in dense subsets of the primes. Let $A$ be a subset of the prime lattice - the d-fold direct product of the primes - of positive relative upper density. We show that A contains all polynomial configurations of the form $x+P_0(y)v_0,\ldots, x+P_l(y)v_l$, for some $x$ in $\mathbb{Z}^d$ and $y$ in $\mathbb{N}$, which satisfy a certain non-degeneracy condition. We also obtain quantitative bounds on the size of such polynomial configuration, if $A$ is a subset of the first $N$ positive integers. |
| title | Polynomial configurations in dense subsets of the prime lattice |
| topic | Number Theory 11N05 |
| url | https://arxiv.org/abs/2504.14424 |