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

• 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 (from the repository page)

• [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).

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 2; best known upper bound 2. Values marked with an asterisk have minimal verification. The recorded bounds coincide, so the repository treats the value as determined.

Known upper bounds

Bound	Reference	Comments
$2$	[Ru96]	$\sigma \le \delta^2$. Attributed to [Ru96, Theorem 4.1] in [GGSWT2025, Problem 6.42]; [Bl26] describes it as a consequence of the Plünnecke–Ruzsa inequalities.

Known lower bounds

Bound	Reference	Comments
$\frac{\log(59/17)}{\log(55/17)} = 1.059793\dots$	[FrPi73]	From the $17$-element set $A = \{0,1,2,4,5,9,12,13,14,16,17,21,24,25,26,28,29\}$, which has $\lvert A+A\rvert = 59$ and $\lvert A-A\rvert = 55$ [GGSWT2025, Problem 6.42]. Used as the baseline in [GGSWT2025] and quoted as such by [LiLi26].
$\frac{\log(32/5)}{\log(26/5)} = 1.125944426\dots$	[PeWe13]	Theorem 21 of [PeWe13]: the supremum of $C(Q_j)$ over their family $Q_j$, approached as $j \to \infty$ but not attained. From their Corollary 13, $\lvert Q_j\rvert = 5j+17$, $\lvert Q_j+Q_j\rvert = 32j+63$ and $\lvert Q_j-Q_j\rvert = 26j+61$, whence $\sigma \to 32/5$ and $\delta \to 26/5$. This held the record until [LiLi26].
$\approx 1.1219$	[GGSWT2025]	This constant is Problem 6.42 of [GGSWT2025], where AlphaEvolve found $A = A_1 \cup A_2$ with $A_1 = \{-159,-158,\dots,111\}$ and $A_2$ an explicit $38$-element set, with no human hints. Inferior to the Penman–Wells record.
$2$	[LiLi26]	Solves the problem: $C_{3d} = 2$. Explicit family with $\sigma \gg K$ and $\delta \ll K^{1/2}$ for arbitrarily large $K$; [LiLi26] states it as $C(A_K) > \frac{2K}{K+3}$ for every positive even $K$.