Seven-corner completion sets

Definitions and mathematical statement

For finite $A\subset{\mathbb{Z}}$, $|A|=n$, define its seven-corner completion set \[ S(A)=\{x+a+b+c:x,x+a,x+b,x+c,x+a+b,x+b+c,x+a+c\in A\}, \] with integer $x,a,b,c$ and no original nonzero-step restriction.

Problem 2.9 (T. Gowers). If $|S(A)|<cn$, what structure of $A$ follows? The constant range and proposed structural conclusion are unspecified. B. Green comments that the answer is "no" as written, that $S$ could be dissociated, and that he, Tim and Terry had not found a satisfactory reformulation.

Reported obstruction. Taking all steps zero shows $A\subseteq S(A)$; thus the hypothesis is vacuous for nonempty $A$ if $c\leq1$. A Sidon set has no nontrivial additive quadruple, equivalently its unordered pair-sums are all distinct. The aid proves every Sidon $A$ satisfies $S(A)=A$. In particular \[ A_n=\{1,3,\ldots,3^{n-1}\},\qquad |S(A_n)|=n,\qquad |A_n+A_n|=n(n+1)/2. \] These sets are dissociated (no nonzero relation with coefficients in $\{-1,0,1\}$). For every $c>1$ they satisfy the printed small-completion condition while having maximal pair-sum growth. This refutes a general small-doubling/standard additive-structure inference, rather than every logically possible statement called "structure."

Reported multiplicity repair. Define $C^{\times}(A)$ to count all ordered quadruples $(x,a,b,c)$ satisfying the seven membership conditions with $abc\ne0$. Let $m_z(A)$ count those with missing vertex $z=x+a+b+c$, so $C^{\times}(A)=\sum_{z\in S(A)}m_z(A)$. These definitions disambiguate the summary's rendered $C_x$ notation. Put \[ E(A)=\#\{(u,v,w,t)\in A^4:u+v=w+t\}. \] The aid's corrected-energy bound is \[ C^{\times}(A)\leq n\big[E(A)-(2n^2-n)\big]. \] The subtracted term counts trivial additive quadruples. The nonzero-step restriction is material; it is not part of the original definition of $S$.

If, for a fixed $\kappa>0$, \[ \frac1{|S(A)|}\sum_{z\in S(A)}m_z(A)\geq\kappa n^3, \] then $|S(A)|\geq n$ and the bound forces additive energy at least $\kappa n^3$. The aid invokes Balog--Szemerédi--Gowers to obtain $A'\subset A$ of size at least $c_\kappa n$ with $|A'-A'|\leq C_\kappa|A'|$. This conclusion needs large nonzero-step multiplicity, not merely few distinct missing vertices; exact constants are absent.

The report also computes, for integer $N\geq0$, \[ S(\{0,1,\ldots,N\}) =[-\lfloor N/2\rfloor,\,\lfloor3N/2\rfloor]\cap{\mathbb{Z}}. \] The counterexample and corrected criterion do not supply the universally preferred reformulation sought in Green's comment.

Short English statement

Few distinct seven-corner completions do not force small doubling: Sidon sets already have completion set equal to themselves. The aid gives an explicit powers-of-three example, then shows that a stronger hypothesis on nonzero-step completion multiplicities forces high additive energy and a large small-difference subset. It also computes the exact interval case.

Sources

• Local UnsolvedMath registry, permanent ID $20001109$, catalogue entry; canonical aim-combinatorics-notes.json, notes[233].
• AIM workshop "Recent Trends in Additive Combinatorics," PDF Problem 2.9, T. Gowers, with B. Green's accompanying negative comment and reformulation discussion: https://aimath.org/WWN/additivecomb/additivecomb.pdf. Supplied verification says the PDF was checked; no independent visit is asserted.
• Balog--Szemerédi--Gowers theorem, as named in the research summary; no full reference or parameter bounds supplied.
• Research attempt 1, counterexample, run status refuted, internal rigor proved, low novelty confidence; registry SOLVED-IN-LITERATURE. Explicitly machine-generated, not peer reviewed, and to be independently verified before citation. Original literature field empty.