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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

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Version history

  1. v1Submitted by alexandre ThieryInitial depositCurrentOct 21, 2026