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on Single-set sum-difference exponent ($C_{3d}$)

The statement

For a finite nonempty subset $A$ of an abelian group, write

\sigma(A) := \frac{|A+A|}{|A|}, \qquad \delta(A) := \frac{|A-A|}{|A|}

for the doubling and difference constants. Ruzsa [Ru96] proved $\delta \le \sigma^2$ , and the Plünnecke–Ruzsa inequalities give the converse $\sigma \le \delta^2$ [Bl26].

For $\lvert A\rvert \ge 2$ with $\delta(A) > 1$ , set $C(A) := \log \sigma(A) / \log \delta(A)$ , and define

C_{3d} := \sup_A C(A)

over all such finite $A$ ; equivalently, $C_{3d}$ is the least exponent $\theta$ for which $\sigma(A) \le \delta(A)^{\theta}$ holds for every finite $A$ . The Plünnecke–Ruzsa bound is exactly the assertion $C_{3d} \le 2$ , and the question — open until 2026 — was whether $\sigma \le \delta^{c}$ is possible for some $c < 2$ .

Sums and differences are not interchangeable for a single set, so this is a genuinely one-sided question; the optimality of the exponent $2$ in the companion inequality $\delta \le \sigma^2$ was already known (see below). The problem is now solved: $C_{3d} = 2$ , by [LiLi26]. The value is approached but not attained by the known constructions.

The supremum is the same whether $A$ ranges over finite subsets of $\mathbb{Z}$ or over finite subsets of arbitrary abelian groups, since a finite subset of an abelian group is Freiman-isomorphic to a subset of $\mathbb{Z}$ [Bl26].

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.