bibkey: costa2026smallabelian
authors: Simone Costa; Stefano Della Fiore; Mattia Fontana; Lluís Vena
year: 2026
title: "Graham conjecture on small sets in abelian groups"
doi: 10.48550/arXiv.2603.20961
url: https://arxiv.org/html/2603.20961v1
claim: The corrected k>=3 forward implication of Theorem 2.4 forces a subgroup carrier from restricted pair sums; the printed k>=2 boundary is false.
strata_touched:
  - D5/S3/FiniteGroups/RestrictedPairSumClassification
license: citation-only
triage: anchor

Graham conjecture on small sets in abelian groups

Theorem 2.4 of arXiv:2603.20961v1 is used as the source for the corrected finite restricted-pair-sum classification. The printed k >= 2 boundary is false: in Z/5Z, A = {1,4} has its only distinct-pair sum equal to zero, but {0,1,4} is not a subgroup because 1+1=2.

The formal proof uses finite translation bijectivity. A non-involution forces a distinct inverse pair by translating A.erase c; the resulting inverse pair gives diagonal closure away from two exceptions. At least three elements provide an element outside this inverse pair; its translation period lets the stabilizer recover both exceptional elements. Thus, for finite nonzero A with 3 <= A.card, the proved implication sends restricted sums in insert 0 A to an additive subgroup carrier. The elementary converse is not formalized in this module.

For such A in the original quotient Q, if insert 0 A is not a subgroup carrier, the contrapositive supplies distinct x_a,x_b whose sum lies outside insert 0 A. Replacing x_a,x_b by x_a+x_b decreases the cardinality by one and preserves distinctness, nonzeroness, the total and the same original Q. The literal lattice pullback through the basis map is available, but a covering interval can split the adjacent (a,b) block at its internal cut. The split-block expansion and the universal ordering target with the same original M and coefficient one in the original coordinates therefore remain open. This note records no novelty claim.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2603.20961
• Version: https://arxiv.org/html/2603.20961v1
• Theorem: 2.4