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Statement

Definition (p. 3). A finite subset $S$ of an abelian group is independent if $\sum_{s\in S}k(s)s\ne0$ for every integer-valued function $k$ on $S$ unless all summands $k(s)s$ are $0$; equivalently, the sum $\bigoplus_{s\in S}\langle s\rangle$ is direct. The edge boundary $\partial_S(A)=|\{(a,s)\in A\times S:a+s\notin A\}|$ is defined on p. 1.

Theorem 2 (p. 3). Let $A$ and $S$ be finite, non-empty subsets of an abelian group, with $S$ independent. Put $n=|S|$ and $d=\min\{\operatorname{ord}s:s\in S\}$. If $\partial_S(A)\le(1-\gamma)n|A|$ with a real $\gamma\in(0,1]$, then

$$|A|\ge4^{(1-1/d)\gamma n}.$$

The paper says the statement is to be read in the expected way when some or all elements of $S$ have infinite order, and that when all of them do, the conclusion reads $|A|\ge4^{\gamma n}$ (p. 3).

Sharpness (pp. 1-3). The abstract calls the constant $4$ best possible; the paper does not say which example shows this. In Example 3 on the page of Theorem 1, the case $t=2$ is the box $[0,1]^n\subseteq C_m^n$ ($m>2$) with the standard generating set, which is independent with $d=m$; there $\gamma=1/2$ and $|A|=4^{\gamma n}$, while the theorem there gives only $|A|\ge4^{(1-1/m)\gamma n}$. The paper says Example 2 shows that the coefficient $1-1/d$ is best possible for $d=2$ and cannot be replaced by a number larger than $\log3/\log4\approx0.792$ for $d=3$ (p. 3).

Open questions (p. 10). The paper asks whether, for every generating subset $S$ of a finite abelian group $G$, the hypothesis $\partial_S(A)\le(1-\gamma)n|A|$ with $n=\operatorname{rk}G$ and real $\gamma\in(0,1]$ implies $|A|\ge4^{(1-1/d)\gamma n}$, with $d$ the least order of an element of $S$. It also asks whether the coefficient $1-1/d$ there can be improved, or dropped, when $G$ is homocyclic with $\exp(G)\ge5$; it calls the case $\exp(G)\le4$ settled by Theorem 1 and Example 2.

Source. Vsevolod F. Lev, On Isoperimetric Stability, Discrete Analysis 2018:14, 11 pp., doi:10.19086/da.3699: the definition and Theorem 2 on p. 3, the proof in Section 3 on pp. 6-9, the questions on p. 10. The edition read is identified on the source card.

Read depth. Claims checked: the definition, the statement, the reading for infinite orders and the sharpness remarks were read clause by clause on the printed pages. The proof (pp. 6-9) was read but not checked step by step.

Proof pointer

Pages 6-9. One may assume that $S=\{s_1,\ldots,s_n\}$ generates the group; the general case follows by the coset decomposition used for Corollary 1. Compressing $A$ along each $s_i$ (pushing the part of $A$ in each $\langle s_i\rangle$-coset to an initial segment of that coset) keeps $|A|$ and, by Claims 1 and 2 (p. 7), produces a set compressed with respect to $S$ without increasing $\partial_S(A)$. For a compressed set the boundary in the direction $s_i$ is the number of $\langle s_i\rangle$-cosets that meet $A$ minus the number contained in $A$, equations (6)-(8). With the hypothesis, equation (9) then gives an average weight (number of non-zero coordinates with respect to $S$) of at least $(1-1/d)\gamma n$. Corollary 2 (p. 8), which carries Theorem 3 over to compressed sets through the coordinate map into $\mathbb Z^n$, bounds the same average by $\frac12\log_2|A|$.

Dependencies

Theorem 3, through Corollary 2 of the same paper. The theorem supplies the first estimate of Theorem 4.

Bears on

• Problem 963: every non-zero real has infinite order, so for finite non-empty $A\subset\mathbb R$ and finite non-empty $S\subset\mathbb R\setminus\{0\}$ with $S$ independent (no non-trivial integer relation) and $\partial_S(A)\le(1-\gamma)|S||A|$ for a real $\gamma\in(0,1]$, the theorem, in its infinite-order reading, gives $|A|\ge4^{\gamma|S|}$. Independence is stronger than the dissociativity the problem asks about, and the theorem gives no bound on the problem's $f(n)$.