HEXAGONbeta

Arithmetic progressions with prescribed lp gaps

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

Contributed by Vjekoslav Kovač ↗

Version 1 / Oct 10, 2026 / CC BY 4.0

Abstract

For every n≥3n\ge3 and p∈[1,∞)∖{1,2,…,n−1}p\in[1,\infty)\setminus\{1,2,\ldots,n-1\}, we prove that every measurable subset of Rd\mathbf{R}^d of positive upper Banach density, with d≥2n−2+1d\ge2^{n-2}+1, contains an nn-term arithmetic progression with every sufficiently large prescribed ℓp\ell^p gap.

Provenance statement

The manuscript was generated gradually in three chats with OpenAI's models, from August 28 to September 9, 2026. (Public links to the chats are provided in the manuscript.) The chats were mutually dependent via ChatGPT's internal library. They were initially directed towards the paper by Durcik and the submitter and the recent blueprint draft by van Doorn, Durcik, Roos, Slavikova, and Thiele. Afterwards the chats were guided by rather generic prompts.

Tools used

OpenAI
ChatGPTVersion 5.6 Sol
OpenAI
ChatGPTVersion 6 Astra

References

  1. E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets , Publ. Math. Inst. Hautes Études Sci. 67 (1988), 5--42. https://doi.org/10.1007/BF02699126 doi:10.1007/BF02699126 .DOI
  2. A. Bostan and P. Dumas, Wronskians and linear independence , Amer. Math. Monthly 117 (2010), no. 8, 722--727. https://doi.org/10.4169/000298910X515785 doi:10.4169/000298910X515785 .DOI
  3. J. Bourgain, A Szemerédi type theorem for sets of positive density in Rk R^k , Israel J. Math. 54 (1986), no. 3, 307--316. https://doi.org/10.1007/BF02764959 doi:10.1007/BF02764959 .DOI
  4. B. Cook, Á. Magyar, and M. Pramanik, A Roth-type theorem for dense subsets of Rd R^d , Bull. Lond. Math. Soc. 49 (2017), no. 4, 676--689. https://doi.org/10.1112/blms.12043 doi:10.1112/blms.12043 .DOI
  5. F. van Doorn, P. Durcik, J. Roos, L. Slavíková, and C. Thiele, A blueprint for the formalization of norm-variation of multiple ergodic averages for commuting transformations , preprint, 2026. https://arxiv.org/abs/2608.27321 arXiv:2608.27321 .arXiv
  6. P. Durcik and V. Kovač, Boxes, extended boxes, and sets of positive upper density in the Euclidean space , Math. Proc. Cambridge Philos. Soc. 171 (2021), no. 3, 481--501. https://doi.org/10.1017/S0305004120000316 doi:10.1017/S0305004120000316 .DOI
  7. P. Durcik and V. Kovač, A Szemerédi-type theorem for subsets of the unit cube , Anal. PDE 15 (2022), no. 2, 507--549. https://doi.org/10.2140/apde.2022.15.507 doi:10.2140/apde.2022.15.507 .DOI
  8. P. Durcik, V. Kovač, and L. Rimanić, On side lengths of corners in positive density subsets of the Euclidean space , Int. Math. Res. Not. IMRN 2018 (2018), no. 22, 6844--6869. https://doi.org/10.1093/imrn/rnx093 doi:10.1093/imrn/rnx093 .DOI
  9. P. Durcik, V. Kovač, K. A. Škreb, and C. Thiele, Norm variation of ergodic averages with respect to two commuting transformations , Ergodic Theory Dynam. Systems 39 (2019), no. 3, 658--688. https://doi.org/10.1017/etds.2017.48 doi:10.1017/etds.2017.48 .DOI
  10. P. Durcik, L. Slavíková, and C. Thiele, Norm-variation of triple ergodic averages for commuting transformations , Anal. PDE 19 (2026), no. 3, 539--586. https://doi.org/10.2140/apde.2026.19.539 doi:10.2140/apde.2026.19.539 .DOI
  11. T. Eisner and T. Tao, Large values of the Gowers--Host--Kra seminorms , J. Anal. Math. 117 (2012), 133--186. https://arxiv.org/abs/1012.3509 arXiv:1012.3509 .arXiv
  12. K. J. Falconer and J. M. Marstrand, Plane sets with positive density at infinity contain all large distances , Bull. Lond. Math. Soc. 18 (1986), no. 5, 471--474. https://doi.org/10.1112/blms/18.5.471 doi:10.1112/blms/18.5.471 .DOI
  13. H. Federer, Geometric Measure Theory , Grundlehren Math. Wiss. 153 , Springer-Verlag, New York, 1969.
  14. H. Furstenberg, Y. Katznelson, and B. Weiss, Ergodic theory and configurations in sets of positive density , in Mathematics of Ramsey Theory , J. Nešetřil and V. Rödl (eds.), Algorithms Combin. 5 , Springer, Berlin, 1990, 184--198. https://doi.org/10.1007/978-3-642-72905-8_13 doi:10.1007/978-3-642-72905-8_13 .DOI
  15. W. T. Gowers, A new proof of Szemerédi's theorem , Geom. Funct. Anal. 11 (2001), no. 3, 465--588. https://doi.org/10.1007/s00039-001-0332-9 doi:10.1007/s00039-001-0332-9 .DOI
  16. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , Princeton Math. Ser. 43 , Princeton University Press, Princeton, NJ, 1993.
  17. E. Szemerédi, On sets of integers containing no kk elements in arithmetic progression , Acta Arith. 27 (1975), 199--245. https://doi.org/10.4064/aa-27-1-199-245 doi:10.4064/aa-27-1-199-245 .DOI
  18. T. Tao, Norm convergence of multiple ergodic averages for commuting transformations , Ergodic Theory Dynam. Systems 28 (2008), no. 2, 657--688. https://doi.org/10.1017/S0143385708000011 doi:10.1017/S0143385708000011 .DOI
  19. P. Varnavides, On certain sets of positive density , J. Lond. Math. Soc. 34 (1959), no. 3, 358--360. https://doi.org/10.1112/jlms/s1-34.3.358 doi:10.1112/jlms/s1-34.3.358 .DOI

Version history

  1. v1Submitted by Vjekoslav KovačInitial depositCurrentOct 10, 2026