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PreprintCC BY 4.0arXiv

Arithmetic progressions in dense subsets of primitive elements of finite fields

  • Hai-Liang Wu
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Abstract

Let p be a prime and let 𝒫p be the set of primitive elements of 𝔽p. Inspired by the work of Cohen, Oliveira e Silva and Trudgian on consecutive primitive elements, and by the work of Chang on arithmetic progressions in multiplicative subgroups of finite fields, in this paper, using estimates for multiplicative character sums over systems of linear forms together with the relative Szemerédi's theorem of Conlon, Fox and Zhao, we show that, for every fixed integer k≥3 and any fixed number 0<δ≤ 1, each subset A⊆𝒫p with #A≥δ#𝒫p contains a nontrivial k-term arithmetic progression, provided that p is sufficiently large. Also, by applying Behrend's theorem, we construct a large subset A' of 𝒫p with #A'=⌊(#𝒫p)1−ε⌋ such that A' contains no nontrivial k-term arithmetic progressions.

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Cite

BibTeXRIS
BibTeX
@misc{wu2026arithmetic,
  title = {{Arithmetic progressions in dense subsets of primitive elements of finite fields}},
  author = {Hai-Liang Wu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29653},
  eprint = {2609.29653},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29653},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Arithmetic progressions in dense subsets of primitive elements of finite fields
AU  - Hai-Liang Wu
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.29653
UR  - https://arxiv.org/abs/2609.29653
AB  - Let p be a prime and let 𝒫_p be the set of primitive elements of 𝔽_p. Inspired by the work of Cohen, Oliveira e Silva and Trudgian on consecutive primitive elements, and by the work of Chang on arithmetic progressions in multiplicative subgroups of finite fields, in this paper, using estimates for multiplicative character sums over systems of linear forms together with the relative Szemerédi's theorem of Conlon, Fox and Zhao, we show that, for every fixed integer k≥3 and any fixed number 0<δ≤ 1, each subset A⊆𝒫_p with #A≥δ#𝒫_p contains a nontrivial k-term arithmetic progression, provided that p is sufficiently large. Also, by applying Behrend's theorem, we construct a large subset A' of 𝒫_p with #A'=⌊(#𝒫_p)^1−ε⌋ such that A' contains no nontrivial k-term arithmetic progressions.
ER  -