Single-set sum-difference exponent

Description of constant

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

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

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].

Known upper bounds

Known lower bounds

Additional comments and links

• The construction. [Bl26] gives a short digest, in a form cleaner than the original. Let $G,H$ be abelian groups of sizes $L,K$, let $S \subset H$ be a Sidon set of size $\asymp K^{1/2}$, let $B \subseteq G$, and set \(A = (G \times \{0\}) \cup (B \times S).\) Then $\lvert A\rvert \asymp L$ provided $\lvert B\rvert \ll L/K^{1/2}$; the largeness of $A+A$ comes from $\lvert S+S\rvert \asymp K$, giving $\sigma \gg \lvert B+B\rvert\lvert S+S\rvert/L \asymp K$ provided $\lvert B+B\rvert \gg L$; and the trivial $\lvert S-S\rvert \le K$ gives $\delta \ll K^{1/2}$ provided $\lvert B-B\rvert \ll LK^{-1/2}$. Hence $\sigma \gg \delta^2$ up to constants, and $C(A) \to 2$ as $K \to \infty$.

  Everything therefore reduces to finding a fixed $B’ \subseteq G’$ with $B’+B’ = G’$ but $B’-B’ \ne G’$, then setting $B = (B’)^d$ and letting $d \to \infty$. [LiLi26] use \(B' = \{0,1,2,4,5,9\} \subseteq \mathbb{Z}/12\mathbb{Z},\) for which $B’+B’$ is all of $\mathbb{Z}/12\mathbb{Z}$ while $B’-B’$ omits the class $6$. The “twist” by $S$ is what makes the argument insensitive to the size of $\lvert B\rvert$ [Bl26].

  [LiLi26] present a longer, fully explicit version that builds the example inside $\mathbb{Z}$ by a Chinese-remainder construction over base-$12$ digit strings, and tracks explicit constants; [Bl26] observes that both features are avoidable.

• Two normalizations — a caution. [PeWe13, §4] track two quantities: the unnormalized \(f(A) = \frac{\log \lvert A+A\rvert}{\log \lvert A-A\rvert} \qquad\text{and}\qquad g(A) = \frac{\log(\lvert A+A\rvert/\lvert A\rvert)}{\log(\lvert A-A\rvert/\lvert A\rvert)},\) the latter being the $C(A)$ used on this page. The classical bounds differ accordingly: $3/4 \le f \le 4/3$ [FrPi73] and $1/2 \le g \le 2$ [Gra06], and so do their records — their Theorem 20 gives $f(Q_{10}) \simeq 1.030597781$, while their Theorem 21 gives $\sup_j g(Q_j) = \log(32/5)/\log(26/5) \simeq 1.125944426$. [Bl26] quotes the previous record as $\sigma \ge \delta^{1.0305\dots}$, which mixes the two: $1.0305\dots$ is the $f$-value, whereas the exponent in $\sigma \ge \delta^{c}$ is the $g$-value $1.125944\dots$ reported by [LiLi26] and recorded in the table above.

• The companion direction. That the exponent $2$ in $\delta \le \sigma^2$ is optimal was classically shown by taking $A$ to be the lattice points of a $d$-dimensional simplex, giving $\sigma \approx 2^d$ and $\delta \approx \binom{2d}{d}$, an example originating in [FrPi73]. The same framework as above does better: taking $B’ = {0,1,3} \subset \mathbb{F}_7$, for which $B’-B’ = \mathbb{F}_7$ but $B’+B’ \ne \mathbb{F}_7$, yields $A$ with $\lvert A-A\rvert \gg K^2\lvert A\rvert$ and $\lvert A+A\rvert \ll K\lvert A\rvert$, hence $\delta \gg \sigma^2$ outright. The simplex example only gives $\delta \gg \sigma^2/\sqrt{\log \sigma}$, so this answers in the negative a question of Ruzsa asking whether $\delta \le \sigma^2$ can be improved by a factor of the shape $(\log \sigma)^c$ [Bl26].

• Generalizations. [Bl26] extends the construction to dilates: if $l_1,l_2,k_1,k_2 \in \mathbb{Z}$ admit a finite $B \subseteq G$ with $l_1B - k_1B = G$ and $l_2B - k_2B \ne G$, then for arbitrarily large $K$ there is $A$ with $\lvert l_1A - k_1A\rvert \gg K^{k_1+l_1}\lvert A\rvert$ and $\lvert l_2A - k_2A\rvert \ll K^{k_2+l_2-1}\lvert A\rvert$. Taking $B’ = {0,1,2} \subset \mathbb{F}_7$ gives sets with $\lvert A+A+A\rvert \gg K^3\lvert A\rvert$ and $\lvert A+A\rvert \ll K\lvert A\rvert$, recovering by a simpler route examples previously constructed by Ruzsa. [Kr26] determines exactly which quadruples $(l_1,l_2,k_1,k_2)$ admit such a $B$: precisely those with $\min(k_2,l_2) > \min(k_1,l_1)$ or $\max(k_2,l_2) > \max(k_1,l_1)$.

• Relation to the other constants in this family. $C_{3a}$ is a two-set sums-versus-differences exponent under a different normalization ($\lvert A+B\rvert \ll \lvert A\rvert$ and $\lvert A-B\rvert \gg \lvert A+B\rvert^{C_{3a}}$); because $B \mapsto -B$ interchanges sums and differences there, it has no analogue of the one-sidedness above, and $C_{3d} = 2$ says nothing directly about $C_{3a} \le 4/3$. $C_{3b}$ and $C_{3c}$ are the Katz–Tao arithmetic-Kakeya sum-difference constants, in which the sumset is restricted to a graph $G \subset A \times B$.

• Thomas Bloom’s note on the construction

References

• [FrPi73] Freiman, G. A.; Pigaev, V. P. The relation between the invariants $r$ and $t$. Kalinin. Gos. Univ., Moscow, 1973, pp. 172–174.
• [Ru96] Ruzsa, I. Z. Sums of finite sets. In: Number Theory: New York Seminar (D. V. Chudnovsky, G. V. Chudnovsky, M. B. Nathanson, eds.), Springer-Verlag, 1996, pp. 281–293.
• [PeWe13] Penman, D.; Wells, M. On sets with more restricted sums than differences. Integers 13 (2013), #A57. The relevant results are Theorems 20 and 21 of §4; see the caution on normalizations above.
• [Gra06] Granville, A. An Introduction to Additive Combinatorics. Lecture notes; cited by [PeWe13] for the bounds $1/2 \le g \le 2$. Published as: CRM Proceedings and Lecture Notes 43, American Mathematical Society, 2007, pp. 1–27.
• [GGSWT2025] Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Mathematical exploration and discovery at scale. arXiv:2511.02864
• [Kr26] Kravitz, N. Inequalities among higher-order difference sets, or, remarks on a construction of Ruzsa. Preprint (2026). arXiv:2606.27087
• [LiLi26] Lin, Haowei; Li, Shanda. Settling the optimal exponent relating sumsets and difference sets. Preprint, 29 July 2026. arXiv:2607.27199. The construction and its proof were developed with the assistance of Hyra, an AI research agent based on the open-weights Hy3 model.
• [Bl26] Bloom, T. F. A sum-difference construction. Note (2026). https://thomasbloom.org/notes/sumdifferences.html (accessed 31 July 2026).

Contribution notes

Prepared with assistance from Claude Opus 5, which read [Bl26] and the arXiv abstract and HTML full text of [LiLi26], and hand-checked the arithmetic of the construction — that $B’+B’ = \mathbb{Z}/12\mathbb{Z}$ while $B’-B’$ omits $6$, that the base-$s$ decomposition used by [LiLi26] is complete, that their restriction to even $K$ is exactly the condition for the relevant moduli to be coprime, and that the resulting asymptotics reproduce the claimed exponent. The exact counting lemmas of [LiLi26] (their Lemmas 2.5–2.7) were not verified, but [Bl26] gives an independent and simpler derivation of the same result. The apparent disagreement between [Bl26] and [LiLi26] over the previous record was resolved against the primary source [PeWe13], which reports both values for different normalizations; the closed form $\log(32/5)/\log(26/5)$ and the cardinalities of $Q_j$ were read off its Theorem 21 and Corollary 13. The bibliographic details for [FrPi73] and [Ru96] follow [Bl26] and have not been checked against the primary sources.