HEXAGONbeta

Distinct partial products in prime dihedral groups

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.CV — Complex Variables

Contributed by alexandre Thiery ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

Let D2pD_{2p} be the dihedral group of order 2p2p, where pp is prime. For every fixed 0<α<10<\alpha<1, we prove that every subset A⊆D2p∖{e}A\subseteq D_{2p}\setminus\{e\} with Cα≤∣A∣≤pαC_\alpha\le |A|\le p^\alpha admits an ordering whose nonempty partial products are pairwise distinct. Combined with the known small- and large-set theorems, this covers every subset size for all sufficiently large primes. Pairing reflections reduces the products to partial sums with fixed parity-labelled offsets. Two passes of block rotations remove distant collisions; bounded displacement allows the local condition to exclude the remaining collisions. The existence of suitable rotations follows from Pham--Sauermann's subset-sum anticoncentration theorem, an affine-rank estimate, and a count of clustered collision witnesses.

Provenance statement

This work began with a reading of Huy Tuan Pham and Lisa Sauermann, *On Graham’s rearrangement conjecture*, arXiv:2602.15797v1, version 1 of 17 February 2026: https://arxiv.org/abs/2602.15797v1. Subsequent discussions with ChatGPT explored how their approach could be adapted to prime dihedral groups. Their subset-sum anticoncentration theorem remains an explicitly cited input to the paper. ChatGPT and Codex were used extensively throughout the development, including proposing and refining proof arguments, checking calculations, investigating simplifications, comparing the construction with prior literature, and preparing the modular LaTeX manuscript. This assistance included substantive work on the cell reduction, pooling and rank estimates, and two-pass repair argument. Separate Codex agents reviewed parts of the proof and their interfaces. Claude supplied a further manuscript audit covering mathematical correctness, exposition, notation and references. Preliminary algebraic components were formalised in Lean, but the complete proof presented in the paper has not been formally verified.

Tools used

OpenAI
ChatGPT
Anthropic
Claude Fable

References

  1. Bapat, R. B. Graphs and Matrices. Springer London. 2010DOI
  2. Benjamin Bedert; Noah Kravitz. Graham's rearrangement conjecture beyond the rectification barrier. Israel Journal of Mathematics, vol. 273, no. 1, pp. 471–500. 2026. Published online 30 November 2025DOI
  3. Benjamin Bedert; Matija Bucić; Noah Kravitz; Richard Montgomery; Alp Müyesser. On Graham's rearrangement conjecture over F2n\mathbb{F}_2^n. 2025. Version 1, 25 August 2025arXiv
  4. Costa, Simone; Della Fiore, Stefano; Engel, Eva R. Graham's rearrangement for a class of semidirect products. 2025. Version 2, 19 August 2025arXiv
  5. Costa, Simone; Della Fiore, Stefano; Feng, Tao; Liu, Hengrui. Kneserized Anticoncentration and Reverse Absorption for Graham's Rearrangement Conjecture. 2026. Version 2, 18 August 2026arXiv
  6. P. Erdős; R. L. Graham. Old and new problems and results in combinatorial number theory. L'Enseignement Mathématique, Université de Genève, no. 28. 1980link
  7. R. L. Graham. On sums of integers taken from a fixed sequence. Proceedings of the Washington State University Conference on Number Theory, pp. 22–40. 1971link
  8. David G. Harris; Aravind Srinivasan. A Constructive Lovász Local Lemma for Permutations. Theory of Computing, vol. 13, no. 17, pp. 1–41. 2017DOI
  9. Kwan, Matthew; Sauermann, Lisa. Resolution of the quadratic Littlewood–Offord problem. Compositio Mathematica, vol. 161, no. 12, pp. 3089–3139. 2025. Lemma 3.7 is cited from arXiv:2312.13826v1arXivDOI
  10. Linyuan Lu; László Székely. Using Lovász Local Lemma in the Space of Random Injections. The Electronic Journal of Combinatorics, vol. 14, no. 1, pp. R63. 2007. Research Paper R63DOI
Show all 13 referencesShow fewer references
  1. Odlyzko, A. M. On subspaces spanned by random selections of ±1\pm1 vectors. Journal of Combinatorial Theory, Series A, vol. 47, no. 1, pp. 124–133. 1988. Proof of Proposition 2.2 is cited from the author's manuscript, pp. 5–6DOI
  2. M. A. Ollis. Sequences in Dihedral Groups with Distinct Partial Products. Australasian Journal of Combinatorics, vol. 78, no. 1, pp. 35–60. 2020link
  3. Huy Tuan Pham; Lisa Sauermann. On Graham's rearrangement conjecture. 2026. Version 1, 17 February 2026arXiv

Version history

  1. v1 · 16 pagesSubmitted by alexandre ThieryInitial depositCurrentOct 21, 2026