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PreprintCC BY 4.0arXiv

Density regularity of {x,x+y,xy} in the integers

  • Florian K. Richter
  • Konstantinos Tsinas
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Abstract

Fix k,s∈ℕ. We prove that there exists a subadditive density on ℕ such that, for every polynomial P∈ℤ[y] of degree k and with P(0)=0, every set of positive density contains configurations {x,x+P(y),xys} for arbitrarily large x>y≥ 2. This provides a density strengthening of Moreira's result on partition regularity for {x,x+y,xy}.

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Cite

BibTeXRIS
BibTeX
@misc{richter2026density,
  title = {{Density regularity of $\x,x+y,xy\$ in the integers}},
  author = {Florian K. Richter and Konstantinos Tsinas},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.30410},
  eprint = {2609.30410},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30410},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Density regularity of {x,x+y,xy} in the integers
AU  - Florian K. Richter
AU  - Konstantinos Tsinas
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.30410
UR  - https://arxiv.org/abs/2609.30410
AB  - Fix k,s∈ℕ. We prove that there exists a subadditive density on ℕ such that, for every polynomial P∈ℤ[y] of degree k and with P(0)=0, every set of positive density contains configurations {x,x+P(y),xy^s} for arbitrarily large x>y≥ 2. This provides a density strengthening of Moreira's result on partition regularity for {x,x+y,xy}.
ER  -