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A Uniform Product-Difference Theorem for Dense Subsets of ℤ2

  • Sayan Goswami
  • Chunlin Liu
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Abstract

We establish a uniform product-difference theorem for dense subsets of ℤ2, which gives an affirmative answer to Problem~2 of Fish and, as consequences, to both parts of his Problem~1. More precisely, we prove that for every δ>0 there exists an integer K(δ)≥ 1 such that every set E⊆ℤ2 with upper Banach density d⋆(E)≥δ satisfies K(δ)ℤ ⊆ {ab:(a,b)∈ E−E}. As consequences, we obtain affirmative answers to both parts of Fish's Problem~1: for positive-density sets E1,E2⊆ℤ and E⊆ℤ2, respectively, the sets (E1−E1)2−(E2−E2)2 and {x2−y2:(x,y)∈ E−E} contain nontrivial ideals of ℤ, with generators depending only on the corresponding density thresholds. In particular, the latter result also settles a conjecture of Davies concerning differences of the indefinite quadratic form x2−y2 in dense subsets of ℤ2.

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Cite

BibTeXRIS
BibTeX
@misc{goswami2026uniform,
  title = {{A Uniform Product-Difference Theorem for Dense Subsets of $\mathbb Z^2$}},
  author = {Sayan Goswami and Chunlin Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06782},
  eprint = {2609.06782},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06782},
}
RIS
TY  - UNPB
TI  - A Uniform Product-Difference Theorem for Dense Subsets of ℤ^2
AU  - Sayan Goswami
AU  - Chunlin Liu
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06782
UR  - https://arxiv.org/abs/2609.06782
AB  - We establish a uniform product-difference theorem for dense subsets of ℤ^2, which gives an affirmative answer to Problem~2 of Fish and, as consequences, to both parts of his Problem~1. More precisely, we prove that for every δ>0 there exists an integer K(δ)≥ 1 such that every set E⊆ℤ^2 with upper Banach density d^⋆(E)≥δ satisfies K(δ)ℤ ⊆ {ab:(a,b)∈ E−E}. As consequences, we obtain affirmative answers to both parts of Fish's Problem~1: for positive-density sets E_1,E_2⊆ℤ and E⊆ℤ^2, respectively, the sets (E_1−E_1)^2−(E_2−E_2)^2 and {x^2−y^2:(x,y)∈ E−E} contain nontrivial ideals of ℤ, with generators depending only on the corresponding density thresholds. In particular, the latter result also settles a conjecture of Davies concerning differences of the indefinite quadratic form x^2−y^2 in dense subsets of ℤ^2.
ER  -