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Triangles in sets of positive upper density in the plane

Primarily AI-generated textHuman understanding: some partsmath.CA — Classical Analysis and ODEsmath.CO — Combinatorics

Contributed by Polona Durcik ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

This note proves that every measurable subset of the plane with positive upper Banach density contains a translate of a rotation of every sufficiently large dilate of any fixed noncollinear triangle.

Provenance statement

The manuscript was generated over three conversations with GPT-6 Astra Pro via Codex on November 4–5, 2026. The model was directed by the submitter toward trilinear oscillatory integral methods, as in the supplied papers by Christ and by Christ, Roos, and the submitter. The rest of the discussion was guided by generic prompts, with the final conversation leading to a substantially shorter argument. The Lean formalization was generated by Claude Opus 5.5 with medium reasoning effort.

Formalizations

Tools used

OpenAI
CodexVersion GPT-6 Astra Pro
Anthropic
Claude CodeVersion Opus 5.5 Medium

References

  1. J. Bourgain, https://doi.org/10.1007/BF02764959 A Szemerédi type theorem for sets of positive density in Rk\R^k , Israel J. Math. 54 (1986), 307--316.DOI
  2. D. Castro-Silva, Geometrical sets with forbidden configurations (https://doi.org/10.1017/fms.2023.43), Forum Math. Sigma 11 (2023), e44.DOI
  3. M. Christ, On certain trilinear oscillatory integral inequalities (https://doi.org/10.1515/ans-2023-0181), Adv. Nonlinear Stud. 25 (2025), 502--553.DOI
  4. M. Christ, A three term sublevel set inequality (https://arxiv.org/abs/2204.04346), arXiv:2204.04346 (2022).arXiv
  5. M. Christ, P. Durcik, and J. Roos, Trilinear smoothing inequalities and a variant of the triangular Hilbert transform (https://doi.org/10.1016/j.aim.2021.107863), Adv. Math. 390 (2021), 107863.DOI
  6. K. Falconer, V. Kovač, and A. Yavicoli, The density of sets containing large similar copies of finite sets (https://doi.org/10.1007/s11854-022-0231-6), J. Anal. Math. 148 (2022), 339--359.DOI
  7. H. Furstenberg and Y. Katznelson, A density version of the Hales--Jewett theorem (https://doi.org/10.1007/BF03041066), J. Anal. Math. 57 (1991), 64--119.DOI
  8. H. Furstenberg, Y. Katznelson, and B. Weiss, Ergodic theory and configurations in sets of positive density (https://doi.org/10.1007/978-3-642-72905-8_13), in Mathematics of Ramsey Theory , Algorithms Combin. 5 , Springer, Berlin, 1990, 184--198.DOI
  9. A. Greenleaf, A. Iosevich, B. Krause, and A. Liu, https://doi.org/10.1007/978-3-031-10796-2_9 LpL^p estimates for bilinear generalized Radon transforms in the plane , in Combinatorial and Additive Number Theory V , Springer Proc. Math. Stat. 395 , Springer, Cham, 2022, 179--198.DOI
  10. T. Ziegler, https://doi.org/10.1007/BF02789447 Nilfactors of Rm\R^m-actions and configurations in sets of positive upper density in Rm\R^m , J. Anal. Math. 99 (2006), 249--266.DOI

Version history

  1. v1 · 9 pagesSubmitted by Polona DurcikInitial depositCurrentNov 11, 2026