The Gyamarti-Hennecart-Ruzsa sum-difference constant
Description of constant
$C_{3a}$ is the largest constant such that there exist arbitrarily large sets $A,B$ of integers such that
\(|A+B| \ll |A|\)
and
\(|A-B| \gg |A+B|^{C_{3a}}.\)
Known upper bounds
| Bound |
Reference |
Comments |
| $4/3 = 1.333\dots$ |
[GHR2007] |
|
Known lower bounds
| Bound |
Reference |
Comments |
| $1$ |
Trivial |
|
| $1.14465$ |
[GHR2007] |
|
| $1.1479$ |
[GGSWT2025] |
AlphaEvolve |
| $1.173050$ |
[G2025] |
|
| $1.173077$ |
[Z2025] |
|
- Thanks to a lemma from [GHR2007], any finite set $U$ of non-negative integers containing zero yields a lower bound of the form
$C_{3a} \geq 1 + \log( \lvert U-U \rvert /\lvert U+U \rvert )/\log(2 \max(U)+1)$. However, the lower bounds obtained in this fashion cannot exceed $1.25$.
- AlphaEvolve repository page for this problem
References
- [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.
- [Z2025] Zheng, Fan. Sums and differences of sets: a further improvement over AlphaEvolve, 2025. arXiv:2506.01896.