Least sizes of non-unique-product sets: the Promislow group and a Heisenberg-type candidate
Abstract
Let P be the Promislow group, the orientable Hantzsche-Wendt group of dimension 3, which underlies Promislow's non-unique-product set and Gardam's counterexample to the unit conjecture. A finite subset A of a group is non-UP if every element of A.A has at least two representations ab with a, b in A. Working in an exact integer model of P, we determine the least size of a non-UP set inside word-balls of the standard generators: it is 14 for every radius from 3 to 6, so a smaller non-UP set of P, if one exists, is not contained in the radius-6 ball. The non-existence half of this statement is certified by machine-checked DRAT and VeriPB proofs. Inside the radius-3 ball there are exactly 16 minimal witnesses, all of point-group distribution (2,6,0,6) up to the swap symmetry. The symmetric non-UP property is not translation invariant, so ball searches cannot be recentred; instead we prove an effective finite-diameter principle: if P contains a non-UP n-set, it contains one inside the ball of explicit radius D(n) = 24(n+1)3^n + 10, so the minimum non-UP cardinality of P is computable in principle. Writing rho(n) for the least word-radius of a non-UP n-set, re-realization experiments on witnesses lead us to conjecture rho(n) = O(n^{1/3}); the bound rho(n) <= 6 (whenever finite) for 8 <= n <= 13 would already show, by the radius-6 computation, that Promislow's 14 is that minimum. We also compute, over balls, the two-sided minimum min(|A|+|B|), the profile beta(m) and the unique-product staircase u(n), and compare with the Fibonacci group H_4 = F(3,4): its least symmetric witness over the radius-4 ball has exactly 16 elements, and its two-sided minimum over the radius-3 ball is 22. Over the stated balls the two groups are ordered oppositely by the symmetric and two-sided invariants (14 < 16 but 24 > 22).
Related
- Unique equivariant pseudo expectation and generalized Powers averaging for crossed product
- Randomized average block coordinate descent method with heavy-ball momentum for tensor least squares problem under the t-product
- Localizing the Gardam unit: the support geometry of units in F_2[P] and its non-unique-product relatives
- The quantitative non-unique-product landscape at the global minimum: the Nielsen-Soelberg groups
- A compactness theorem for twisted-unitary elements of integral group rings, with a certified route to the theta-unitary Case A at window B(4) of the Promislow group
BibTeX
@misc{tabei2026least,
title = {{Least sizes of non-unique-product sets: the Promislow group and a Heisenberg-type candidate}},
author = {Moe Tabei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.18346},
eprint = {2607.18346},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.18346},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Least sizes of non-unique-product sets: the Promislow group and a Heisenberg-type candidate
AU - Moe Tabei
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2607.18346
UR - https://arxiv.org/abs/2607.18346
AB - Let P be the Promislow group, the orientable Hantzsche-Wendt group of dimension 3, which underlies Promislow's non-unique-product set and Gardam's counterexample to the unit conjecture. A finite subset A of a group is non-UP if every element of A.A has at least two representations ab with a, b in A. Working in an exact integer model of P, we determine the least size of a non-UP set inside word-balls of the standard generators: it is 14 for every radius from 3 to 6, so a smaller non-UP set of P, if one exists, is not contained in the radius-6 ball. The non-existence half of this statement is certified by machine-checked DRAT and VeriPB proofs. Inside the radius-3 ball there are exactly 16 minimal witnesses, all of point-group distribution (2,6,0,6) up to the swap symmetry. The symmetric non-UP property is not translation invariant, so ball searches cannot be recentred; instead we prove an effective finite-diameter principle: if P contains a non-UP n-set, it contains one inside the ball of explicit radius D(n) = 24(n+1)3^n + 10, so the minimum non-UP cardinality of P is computable in principle. Writing rho(n) for the least word-radius of a non-UP n-set, re-realization experiments on witnesses lead us to conjecture rho(n) = O(n^{1/3}); the bound rho(n) <= 6 (whenever finite) for 8 <= n <= 13 would already show, by the radius-6 computation, that Promislow's 14 is that minimum. We also compute, over balls, the two-sided minimum min(|A|+|B|), the profile beta(m) and the unique-product staircase u(n), and compare with the Fibonacci group H_4 = F(3,4): its least symmetric witness over the radius-4 ball has exactly 16 elements, and its two-sided minimum over the radius-3 ball is 22. Over the stated balls the two groups are ordered oppositely by the symmetric and two-sided invariants (14 < 16 but 24 > 22).
ER -