A Kneser–Pollard theorem for higher representation sums

Let $t$ be a positive integer, let $G$ be an abelian group, and let $A,B\subseteq G$ be finite subsets with $|A|,|B|\geq t$. Write $A+_iB$ for the set of elements of $G$ having at least $i$ representations as a sum from $A+B$, and let $\mathsf H(S)$ denote the stabilizer of a set $S$. If

$$\sum_{i=1}^t |A+_iB|<t|A|+t|B|-t^2,$$

then there exist subsets $A'\subseteq A$ and $B'\subseteq B$ such that

$$|A\setminus A'|+|B\setminus B'|<u,\qquad A'+B'=A'+_uB'=A+_uB,$$

and, on writing

$$H:=\mathsf H(A'+B')=\mathsf H(A'+_uB')=\mathsf H(A+_tB),$$

we have

$$\sum_{i=1}^t|A+_iB|\geq t|A|+t|B|-t^2-u(|H|-u),$$

where $t=s|H|+u$ with $u\in[1,|H|]$ and $s\geq0$ integers. Higher Kneser–Pollard conjecture. The stated structural conclusion should hold under the displayed small-sum hypothesis. The conjecture seeks a common generalization of Pollard's and Kneser's theorems for finite subsets of arbitrary abelian groups. The surrounding discussion presents it as a strong possible formulation and notes that further investigation may require altering portions of the statement; its resolution status is therefore not established by the source.