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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.27581 |
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| _version_ | 1866915588773773312 |
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| author | Fan, Steve Lott, Andrew |
| author_facet | Fan, Steve Lott, Andrew |
| contents | Fix a positive prime power $q$, and let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$. Suppose $A \subseteq \{f \in \mathbb{F}_q[t]\colon\text{deg}~ f \le N\}$ contains no pair of elements whose difference is of the form $P-1$ with $P$ irreducible. Adapting Green's approach to Sárközy's theorem for shifted primes in $\mathbb{Z}$ using the van der Corput property, we show that \[|A| \ll q^{(N+1)(11/12+o(1))},\] improving upon the bound $O\big(q^{(1-c/\log N)(N+1)}\big)$ due to Lê and Spencer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_27581 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sárközy's theorem in $\mathbb{F}_q[t]$ via the van der Corput property Fan, Steve Lott, Andrew Number Theory Fix a positive prime power $q$, and let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$. Suppose $A \subseteq \{f \in \mathbb{F}_q[t]\colon\text{deg}~ f \le N\}$ contains no pair of elements whose difference is of the form $P-1$ with $P$ irreducible. Adapting Green's approach to Sárközy's theorem for shifted primes in $\mathbb{Z}$ using the van der Corput property, we show that \[|A| \ll q^{(N+1)(11/12+o(1))},\] improving upon the bound $O\big(q^{(1-c/\log N)(N+1)}\big)$ due to Lê and Spencer. |
| title | Sárközy's theorem in $\mathbb{F}_q[t]$ via the van der Corput property |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.27581 |