Reasonable bounds for Restricted Polynomial Progressions in 𝔽^n_p

Abstract

Given an integer k ≥ 2 and a prime p large enough depending on k, we show that, if A ⊆ 𝔽ⁿ_p does not contain a restricted polynomial progression of the form x, x+ y₁ y₂ vdots y_n , x+ y₁² y₂² vdots y_n² , …, x+ y₁^k−1 y₂^k−1 vdots y_n^k−1 , where x ∈ 𝔽ⁿ_p and y ∈ {0,2, 3, …, M_k}ⁿ backslash {0⃗ }, for some fixed M_k, then A must have density bounded above by C/((log^{(k^O(k))} n)^c), where C, c>0 only depend on k and p. For k≥ 4, the best previously known bound came from the density Hales--Jewett theorem and was of inverse Ackermann type. The proof uses a recent inverse theorem for noise stability of Bhangale, Khot, Liu and Minzer, which arose in the study of constraint satisfaction problems. The main new ingredient is a proof that the underlying distribution counting restricted polynomial progressions does not admit a nontrivial abelian embedding.

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BibTeX RIS

BibTeX

@misc{burgin2026reasonable,
  title = {{Reasonable bounds for Restricted Polynomial Progressions in $\mathbb F^n_p$}},
  author = {Alex Burgin and Anastasios Fragkos and David Kruzel and Michael Lacey and Amelia Stokolosa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12831},
  eprint = {2610.12831},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12831},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Reasonable bounds for Restricted Polynomial Progressions in 𝔽^n_p
AU  - Alex Burgin
AU  - Anastasios Fragkos
AU  - David Kruzel
AU  - Michael Lacey
AU  - Amelia Stokolosa
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.12831
UR  - https://arxiv.org/abs/2610.12831
AB  - Given an integer k ≥ 2 and a prime p large enough depending on k, we show that, if A ⊆ 𝔽^n_p does not contain a restricted polynomial progression of the form x, x+ y_1 y_2 vdots y_n , x+ y_1^2 y_2^2 vdots y_n^2 , …, x+ y_1^k−1 y_2^k−1 vdots y_n^k−1 , where x ∈ 𝔽^n_p and y ∈ {0,2, 3, …, M_k}^n backslash {0⃗ }, for some fixed M_k, then A must have density bounded above by C/((log^(k^O(k)) n)^c), where C, c>0 only depend on k and p. For k≥ 4, the best previously known bound came from the density Hales--Jewett theorem and was of inverse Ackermann type. The proof uses a recent inverse theorem for noise stability of Bhangale, Khot, Liu and Minzer, which arose in the study of constraint satisfaction problems. The main new ingredient is a proof that the underlying distribution counting restricted polynomial progressions does not admit a nontrivial abelian embedding.
ER  -

Reasonable bounds for Restricted Polynomial Progressions in 𝔽ⁿ_p

Abstract

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