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Statement

Corollary 1 (p. 2). Let $A$ and $S$ be finite, non-empty subsets of an abelian group $G$ with $\exp(G)\in\{2,3\}$. Put $H=\langle S\rangle$ and $n=\operatorname{rk}H$. If $\partial_S(A)\le(1-\gamma)n|A|$ with a real $\gamma\in(0,1]$, then

$$|A|\ge|H|^{\gamma}.$$

Here $\partial_S(A)$ is the edge boundary defined on the page of Theorem 1. The set $S$ need not generate $G$ and need not be independent. The paper notes (p. 2) that Theorem 1 admits no straightforward extension to $\exp(G)>4$ (Example 3).

Source. Vsevolod F. Lev, On Isoperimetric Stability, Discrete Analysis 2018:14, 11 pp., doi:10.19086/da.3699: Corollary 1 on p. 2, its proof on p. 6. The edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The proof (p. 6) was read but not checked step by step.

Proof pointer

Page 6. The case $S=\{0\}$ is immediate. Otherwise write $A$ as a disjoint union of translates $a_i+A_i$ with $A_i\subseteq H$, one for each $H$-coset that $A$ meets. The boundary $\partial_S$ adds over these pieces, so some $A_i$ satisfies $\partial_S(A_i)\le(1-\gamma)n|A_i|$. As $\exp(H)\in\{2,3\}$, the group $H$ is homocyclic and generated by $S$, and Theorem 1 gives $|A|\ge|A_i|\ge|H|^\gamma$.

Dependencies

Theorem 1 of the same paper. The corollary supplies the exponent-3 estimate of Theorem 4.