Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-d inequality

Abstract

Green and Sawhney recently obtained a quasipolynomial bound in the Furstenberg--Sárközy theorem for square differences by proving an ``arithmetic level-d'' inequality, thereby yielding a greatly improved density increment scheme. We apply their method to treat general intersective polynomials h∈ℤ[x]. In particular, let D(h(ℕ),X):= max|A|: A⊆ [1,X]∩ℕ and (A−A)∩ h(ℕ)⊆{0}. We prove that for every 0<μ<1/2 there are constants c₀, X_min>0 depending on h and μ such that for every X>X_min, D(h(ℕ), X)≤ Xe^{−c₀(log X)^μ}. This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences, improving upon the work of Arala. In order to achieve the admissible exponent range 0<μ<1/2, we use sieve methods to develop novel exponential sum estimates in the style of Rice, and we use the ``random sparsification'' procedure of Green and Sawhney.

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BibTeX RIS

BibTeX

@misc{adajar2026extensions,
  title = {{Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-$d$ inequality}},
  author = {Carlo Francisco E. Adajar and Rishika Agrawal and Mukul Rai Choudhuri and Chian Yeong Chuah and Steve Fan and Swaroop Hegde and Andrew Lott and Krishnamohan Nandakumar and Nagendar Reddy Ponagandla},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2605.16216},
  eprint = {2605.16216},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.16216},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-d inequality
AU  - Carlo Francisco E. Adajar
AU  - Rishika Agrawal
AU  - Mukul Rai Choudhuri
AU  - Chian Yeong Chuah
AU  - Steve Fan
AU  - Swaroop Hegde
AU  - Andrew Lott
AU  - Krishnamohan Nandakumar
AU  - Nagendar Reddy Ponagandla
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2605.16216
UR  - https://arxiv.org/abs/2605.16216
AB  - Green and Sawhney recently obtained a quasipolynomial bound in the Furstenberg--Sárközy theorem for square differences by proving an ``arithmetic level-d'' inequality, thereby yielding a greatly improved density increment scheme. We apply their method to treat general intersective polynomials h∈ℤ[x]. In particular, let D(h(ℕ),X):= max|A|: A⊆ [1,X]∩ℕ and (A−A)∩ h(ℕ)⊆{0}. We prove that for every 0<μ<1/2 there are constants c_0, X_min>0 depending on h and μ such that for every X>X_min, D(h(ℕ), X)≤ Xe^−c_0(log X)^μ. This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences, improving upon the work of Arala. In order to achieve the admissible exponent range 0<μ<1/2, we use sieve methods to develop novel exponential sum estimates in the style of Rice, and we use the ``random sparsification'' procedure of Green and Sawhney.
ER  -

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