Gyarmati-Hennecart-Ruzsa sum-difference constant ($C_{3a}$)

• A lemma from [GHR2007] states that a finite set $U$ of non-negative integers containing zero and satisfying $\lvert U-U\rvert<2\max(U)+1$ yields $C_{3a} \geq 1 + \log( \lvert U-U \rvert /\lvert U+U \rvert )/\log(2 \max(U)+1)$. Lower bounds obtained in this fashion cannot exceed $1.25$.
• Certified values and limits (the asterisked rows). As set out in CONTRIBUTING.md, the Bound column is meant to hold values that a reader can recompute directly. The rows marked $\*$ are instead limits: each is the supremum of a sequence of per-depth lemma values $\theta(U_d)$, every one of which is itself a valid lower bound, so the limit is a valid lower bound too — but none of them is certified by a finite-depth computation, and each rests on the asymptotic analysis in its cited source. The largest value here certified by exact finite counting is $1.1835129324$ [MI2026]; per [Num2026], applying the limit principle to that same base-$33$ alphabet would already give $1.18552$.
• Reproducing the $1.14465$ of [GHR2007]. The final construction of [GHR2007, §2] is recorded there with $\lvert U+U\rvert = 4455634$, $\lvert U-U\rvert = 110205905$ and $q = 2\max U + 1 = 5723906483$, and the exponent is stated as $\theta = 1.144655$ (Theorem 1 asserts $\theta_0 > 1.14465$). Substituting those three values into the lemma gives $1 + \log(110205905/4455634)/\log(5723906483) = 1.142789\dots$, which is smaller. The same substitution reproduces the paper's other stated exponents to six decimal places ($1.1078\dots$, $1.1165757\dots$ and $1.135589\dots$ against a printed $1.135596$), so the method of checking appears sound; the printed multiplier sequence $\Lambda$ for $m = 11$ may not be the one that produced the record. All later entries in the table exceed $1.14465$ regardless, so nothing downstream depends on this.
• [K2026] generalizes the bounded-digit limit construction in [Z2025] by replacing bounded digits with digits restricted to a finite mask.
• The record mask in [K2026] is generated by the product grid $\{3,4\}\times\{506,508,529\}$; its column semigroup is the simple gluing $\langle506,508,529\rangle=23\langle22,23\rangle+508\mathbb N_0$.
• [K2026b] generalizes [K2026] and [Z2025] by allowing constructions in arbitrary bases greater than the maximum element of the digit mask and accounting for carrying. The record mask is different from that of [K2026], but the semigroup generators again form a product grid.
• AlphaEvolve repository page for this problem

References (from the repository page)

• [GGSWT2025] Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Mathematical exploration and discovery at scale. arXiv:2511.02864
• [G2025] Gerbicz, Robert. Sums and differences of sets (improvement over AlphaEvolve), 2025. arXiv:2505.16105.
• [GHR2007] Gyarmati, Katalin; Hennecart, François; Ruzsa, Imre Z. Sums and differences of finite sets. Functiones et Approximatio Commentarii Mathematici, 37(1):175–186, 2007.
• [HRY1999] Hennecart, François; Robert, Gilles; Yudin, Alexander. On the number of sums and differences. In: Structure Theory of Set Addition, Astérisque 258 (1999), 173–178.
• [Ru96] Ruzsa, Imre Z. Sums of finite sets. In: Number Theory (New York, 1991–1995), Springer, New York, 1996, pp. 281–293.
• [MI2026] Mosaic Intelligence (@111111). Exact-count certificate for problem 3a, certificate archive, submitted to this repository (2026).
• [Num2026] Numaro (numaro.tech). Large-deviation limit certificate for a base-89 capped digit construction, certificate archive, submitted to this repository (2026).
• [Z2025] Zheng, Fan. Sums and differences of sets: a further improvement over AlphaEvolve, 2025. arXiv:2506.01896.
• [G2026] Griego, Sebastian. Base-$21$ digit construction certificate for $C_{3a}$, submitted to this repository (2026).
• [K2026] Kleinwaks, Logan. A masked-digit lower bound for the Gyarmati–Hennecart–Ruzsa sum–difference constant, proof and verification package, submitted to this repository (2026).
• [K2026b] Kleinwaks, Logan. Improved lower bound for the Gyarmati–Hennecart–Ruzsa sum–difference constant using masked digits and controlled carries, proof, verification package, and Lean formalization, submitted to this repository (2026).

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1.19102809 (1.19519192*); best known upper bound 1.33333. Values marked with an asterisk have minimal verification.

Known upper bounds

Known lower bounds