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Statement

Notation (p. 4). For a finite subset $A$ of an abelian group $G$, $\dim_I(A)$ is the largest size of an independent subset of $A$ (independence as defined on the page of Theorem 2). For $g\in G$, $r_A(g)=|\{(a,a')\in A\times A:g=a-a'\}|$, and for real $\gamma\in(0,1]$ the set of $\gamma$-popular differences is $P_\gamma(A)=\{g\in G:r_A(g)\ge\gamma|A|\}$.

Theorem 4 (p. 4). Let $p$ be the smallest order of a non-zero element of an abelian group $G$. For every finite, non-empty $A\subseteq G$ and every real $\gamma\in[0,1)$,

$$\dim_I(P_\gamma(A))\le(2(1-1/p))^{-1}\gamma^{-1}\log_2|A|.$$

If moreover $\exp(G)=3$, then

$$\dim_I(P_\gamma(A))\le\gamma^{-1}\log_3|A|.$$

The range $\gamma\in[0,1)$ is as printed in the theorem, while $P_\gamma(A)$ was defined just above it for $\gamma\in(0,1]$; at $\gamma=0$ the right-hand sides are not finite numbers. Read literally, the definition of independence on p. 3 also admits sets containing $0$ (for instance $\{0\}$, so that $\dim_I(P_\gamma(\{0\}))=1$ while the right-hand sides vanish at $A=\{0\}$); the proof on p. 10 applies Theorem 2 with every element of $S$ of order at least $p$, which holds for independent sets $S$ not containing $0$.

Sharpness (p. 4). The paper says the estimate is sharp for $G$ homocyclic of exponent $m\in\{2,3\}$ and reasonably close to sharp for $m\ge4$. Example 4: for integers $m\ge2$ and $k,n\ge1$ with $k\mid n$, write $C_m^n=H_1\oplus\cdots\oplus H_k$ with each $H_i\cong C_m^{n/k}$ and put $A=H_1\cup\cdots\cup H_k$. Then $|A|=k(m^{n/k}-1)+1\le km^{n/k}$, every non-zero $a\in A$ has $r_A(a)=m^{n/k}$, and with $\gamma=m^{n/k}/|A|\ge k^{-1}$

$$\dim_I(P_\gamma(A))\ge n=k\log_m(\gamma|A|)\ge\gamma^{-1}\log_m|A|-\gamma^{-1}\log_m(\gamma^{-1}).$$

Dissociated sets (pp. 4-5, 10). A subset $A$ is dissociated if the sums $\sum_{a\in B}a$, $B\subseteq A$, are pairwise distinct, and the additive dimension $\dim_D(A)$ is the size of its largest dissociated subset. The paper reads Shkredov and Yekhanin [SY11, Theorem 3.1] as saying essentially that for $A$ in a finite abelian group, $\dim_D(P_\gamma(A))\ll\gamma^{-1}\log|A|$ with an absolute implicit constant, its (1). Every independent set is dissociated, and the two notions coincide in groups of exponent $2$ or $3$, so $\dim_I(P)\le\dim_D(P)$ for every subset $P$, with equality in exponent $2$ or $3$. The paper concludes that (1) is qualitatively stronger than Theorem 4 for groups of exponent larger than $3$, while Theorem 4 is stronger than (1) in exponents $2$ and $3$, where its coefficients are sharp.

A remark credited to Thomas Bloom (personal communication) at the start of Section 5 (p. 10) states that if $S$ is dissociated, then $\partial_S(A)\le(1-\gamma)|A||S|$ implies $|A|>\exp(c\gamma^2|S|)$ with an absolute constant $c>0$. The paper indicates that it follows from Hölder's inequality and basic Fourier analysis, gives no proof, and states no further hypothesis on the group beyond the abelian group $G$ of its setting. The paper asks for the best possible coefficient in Theorem 4 for homocyclic groups of exponent larger than $3$ (p. 11).

Source. Vsevolod F. Lev, On Isoperimetric Stability, Discrete Analysis 2018:14, 11 pp., doi:10.19086/da.3699: the notation, Theorem 4, Example 4 and the comparison with [SY11] (I. Shkredov and S. Yekhanin, J. Combin. Theory Ser. A 118 (2011), 1086-1093) on pp. 4-5, the proof in Section 4 on p. 10, the remark and question in Section 5 on pp. 10-11. The edition read is identified on the source card.

Read depth. Claims checked: the notation, the statement, Example 4, the comparison with (1) and the Section 5 remark were read clause by clause on the printed pages. The proof (p. 10) was read but not checked step by step. The Section 5 remark has no proof in the paper and none was checked here.

Proof pointer

Page 10. If $S\subseteq P_\gamma(A)$ is independent with $|S|=n$, each $s\in S$ has at least $\gamma|A|$ representations $s=a'-a$, so at least $\gamma|A|n$ pairs $(a,s)$ have $a+s\in A$ and $\partial_S(A)\le(1-\gamma)n|A|$. Theorem 2 gives $|A|\ge4^{(1-1/p)\gamma n}$, and taking $n=\dim_I(P_\gamma(A))$ gives the first estimate. For $\exp(G)=3$, Corollary 1 in place of Theorem 2 gives $|A|\ge3^{\gamma n}$.

Dependencies

Theorem 2 and Corollary 1 of the same paper.

Bears on

• Problem 963: the problem asks for a lower bound on the largest dissociated subset of every $n$-element set of reals. Theorem 4 bounds independent subsets of a popular-difference set from above. In $\mathbb R$ no non-zero element has finite order; the theorem does not say how $p$ is read there, and the argument of Section 4 with the infinite-order reading of Theorem 2 gives $|S|\le(2\gamma)^{-1}\log_2|B|$ for every independent $S\subseteq P_\gamma(B)\setminus\{0\}$, with $B\subset\mathbb R$ finite and non-empty and $0<\gamma<1$. Over $\mathbb R$ independence (no non-trivial integer relation) is stronger than dissociativity, so this does not bound dissociated subsets. The Section 5 remark concerns dissociated sets but is stated without proof. Neither gives a bound on the problem's $f(n)$.