An odd-order five-coset counterexample above the one-third threshold

Definitions and mathematical statement

Let $A$ be a finite nonempty subset of an abelian group $G$, let $D=A-A$, and count ordered representations by \[ r_A(d)=\#\{(a',a'')\in A^2:a'-a''=d\}. \] V. Lev's source records that $r_A(d)>|A|/2$ for every $d\in D$ forces $D$ to be a subgroup. At the non-strict threshold, the example $A=H\cup(g+H)$, where $H$ is a finite subgroup and $g+H$ has order at least five in $G/H$, has $D=(-g+H)\cup H\cup(g+H)$, not a subgroup, while every represented difference has at least $|H|=|A|/2$ representations.

The source asks two separate questions: whether $r_A(d)\geq|A|/2$ for every $d\in D$ always makes $D$ a subgroup or a union of three cosets of a common subgroup, and whether the same conclusion holds under the weaker strict hypothesis \[ r_A(d)>|A|/3\qquad(d\in D). \] The strictness and the distinction between these two thresholds are essential.

The attached aid refutes the one-third assertion with $G={\mathbb{Z}}/3{\mathbb{Z}}\times{\mathbb{Z}}/7{\mathbb{Z}}$ and \[ A=({\mathbb{Z}}/3{\mathbb{Z}}\times\{0\})\cup(\{0,1\}\times\{1\})\cup({\mathbb{Z}}/3{\mathbb{Z}}\times\{2\}). \] It records $|A|=8$ and $D={\mathbb{Z}}/3{\mathbb{Z}}\times\{0,\pm1,\pm2\}$. On heights $\pm2$ the multiplicity is $3$ for each first coordinate; on heights $\pm1$ it is $4$; on height zero the values at first coordinates $0,1,2$ are respectively $8,7,7$. Thus every $d\in D$ has $r_A(d)\geq3>8/3$. The $15$-element set $D$ is neither a subgroup nor a union of at most three cosets of any common subgroup of the order-$21$ group. This is an odd-order example, so absence of nonzero two-torsion does not rescue the one-third conclusion. Its minimum relative multiplicity $3/8$ does not satisfy the half-threshold hypothesis and therefore does not settle that separate question.

More generally, in the same three-layer construction with full outer fibers of size $h$ and a nonempty middle fiber of size $m\leq h$, the aid records relative minimum $\min(h,2m)/(2h+m)$. This is at most $2/5$, with equality at $m=h/2$ when that integer size is available. This construction-specific ratio is not claimed to be a universal classification threshold.

Short English statement

Ask whether uniformly popular differences force a subgroup or three-coset structure. The supplied eight-element odd-order example refutes the strict one-third version. It does not answer the separate question at multiplicity at least one half.

Sources

• AIM list Recent trends in additive combinatorics, problem 1.18 attributed to V. Lev: https://aimath.org/WWN/additivecomb/additivecomb.pdf.
• Local registry data/unsolvedmath/problems.json, permanent ID 20001099, exact source code catalogue entry; canonical location aim-combinatorics-notes.json notes[223].
• Attached attempt 1, counterexample, problem status at run refuted, rigor proved, low novelty confidence. The explicit example and general three-layer ratio have no separate complete external reference in the supplied record; the aid is machine-generated, not peer reviewed, requiring independent verification before citation.