Furstenberg--Sárközy theorem over number fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916996988272640 |
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| author | Pandey, Dev Ranjan Saha, Jyoti Prakash |
| author_facet | Pandey, Dev Ranjan Saha, Jyoti Prakash |
| contents | We introduce the notion of intersective polynomials having coefficients in the ring of integers $\mathscr{O}_K$ of a number field $K$, and define a notion of upper density of subsets of $\mathscr{O}_K$. We prove that given any intersective polynomial $p(x)$ over $\mathscr{O}_K$, every subset $A$ of $\mathscr{O}_K$ of positive upper density contains two distinct elements whose difference is equal to $p(x)$ for some element $x$ in $\mathscr{O}_K$. Moreover, we obtain a quantitative version of this result. The proof is motivated by an argument due to Lucier, and the Fourier-free proof of the Furstenberg--Sárközy theorem over the integers by Green, Tao and Ziegler. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18990 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Furstenberg--Sárközy theorem over number fields Pandey, Dev Ranjan Saha, Jyoti Prakash Number Theory Combinatorics 11B30, 05B10 We introduce the notion of intersective polynomials having coefficients in the ring of integers $\mathscr{O}_K$ of a number field $K$, and define a notion of upper density of subsets of $\mathscr{O}_K$. We prove that given any intersective polynomial $p(x)$ over $\mathscr{O}_K$, every subset $A$ of $\mathscr{O}_K$ of positive upper density contains two distinct elements whose difference is equal to $p(x)$ for some element $x$ in $\mathscr{O}_K$. Moreover, we obtain a quantitative version of this result. The proof is motivated by an argument due to Lucier, and the Fourier-free proof of the Furstenberg--Sárközy theorem over the integers by Green, Tao and Ziegler. |
| title | Furstenberg--Sárközy theorem over number fields |
| topic | Number Theory Combinatorics 11B30, 05B10 |
| url | https://arxiv.org/abs/2508.18990 |