PreprintCC BY 4.0arXiv
A Generalization of Sárközy's theorem in function fields
Abstract
Sárközy's theorem says that if A ⊂ ℤ has positive upper asymptotic density, then there are distinct a1, a2 ∈ A and n ∈ ℤ such that a1−a2 = n2. The same is true if n2 is replaced by F(n) for any polynomial F ∈ ℤ[x] with constant term zero. Green proved an 𝔽q[t]-analog of Sárközy's theorem with strong quantitative bounds, but required a technical condition on the number of roots of the polynomial F ∈ 𝔽q[x]. This condition was recently removed by Li and Sauermann. In this paper, we generalize Green's argument to accommodate equations in more variables in 𝔽q[t], while pointing out that the technical condition can be removed by means of a simple observation.
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BibTeX
@misc{bienvenu2026generalization,
title = {{A Generalization of Sárközy's theorem in function fields}},
author = {Pierre-Yves Bienvenu and Thái Hoàng Lê and Gauree Wathodkar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12499},
eprint = {2609.12499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12499},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Generalization of Sárközy's theorem in function fields
AU - Pierre-Yves Bienvenu
AU - Thái Hoàng Lê
AU - Gauree Wathodkar
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12499
UR - https://arxiv.org/abs/2609.12499
AB - Sárközy's theorem says that if A ⊂ ℤ has positive upper asymptotic density, then there are distinct a_1, a_2 ∈ A and n ∈ ℤ such that a_1−a_2 = n^2. The same is true if n^2 is replaced by F(n) for any polynomial F ∈ ℤ[x] with constant term zero. Green proved an 𝔽_q[t]-analog of Sárközy's theorem with strong quantitative bounds, but required a technical condition on the number of roots of the polynomial F ∈ 𝔽_q[x]. This condition was recently removed by Li and Sauermann. In this paper, we generalize Green's argument to accommodate equations in more variables in 𝔽_q[t], while pointing out that the technical condition can be removed by means of a simple observation.
ER -