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Improve C_3a lower bound to 1.19102809 via masked semigroup digits #134

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Improve C_3a lower bound to 1.19102809 via masked semigroup digits #134

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1360842
Improve lower bound for C_3a to 1.19023813
kleinwaks Jul 24, 2026
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Update comment about mask in [K2026] generalization of [Z2025]
kleinwaks Jul 24, 2026
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Update [K2026] reference with pull request URL
kleinwaks Jul 24, 2026
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Update C_3a record to 1.19102809
kleinwaks Jul 26, 2026
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Fix markdown in 3a.md
kleinwaks Jul 26, 2026
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18 changes: 11 additions & 7 deletions constants/3a.md
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Original file line number Diff line number Diff line change
@@ -1,4 +1,4 @@
# The Gyamarti-Hennecart-Ruzsa sum-difference constant
# The Gyarmati-Hennecart-Ruzsa sum-difference constant

## Description of constant

Expand All @@ -18,18 +18,21 @@ $$ |A-B| \gg |A+B|^{C_{3a}}.$$
| Bound | Reference | Comments |
| ----- | --------- | -------- |
| $1$ | Trivial | |
| $1.14465$ | [GHR2007] |
| $1.1479$ | [GGSWT2025] | AlphaEvolve
| $1.173050$ |[G2025] |
| $1.173077$ |[Z2025]|
| $1.14465$ | [GHR2007] | |
| $1.1479$ | [GGSWT2025] | AlphaEvolve |
| $1.173050$ |[G2025] | |
| $1.173077$ |[Z2025]| |
| $1.1740744$ | [G2026] | Base-$21$ digit construction with exact counting certificate. |
| $1.1835129324$ | [MI2026] | Base-$33$ digit construction with exact counting certificate. |
| $1.19102809$ | [K2026] | Base-$34065$ masked-digit limit construction with $M=\langle1518,1524,1587,2024,2032,2116\rangle\cap[0,17032]$ and a directed-rounding certificate. |


## Additional comments and links

- 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 $1.25$.
- 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 $1.25$.
- [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 $\langle506,508,529\rangle=23\langle22,23\rangle+508\mathbb N_0$.
- [AlphaEvolve repository page for this problem](https://google-deepmind.github.io/alphaevolve_repository_of_problems/problems/44.html)

## References
Expand All @@ -40,3 +43,4 @@ $C_{3a} \geq 1 + \log( \lvert U-U \rvert /\lvert U+U \rvert )/\log(2 \max(U)+1)$
- [MI2026] Mosaic Intelligence ([@111111](https://x.com/111111)). Exact-count certificate for problem 3a, [certificate archive](https://doi.org/10.5281/zenodo.20794135), [submitted to this repository](https://github.com/teorth/optimizationproblems/pull/95) (2026).
- [Z2025] Zheng, Fan. Sums and differences of sets: a further improvement over AlphaEvolve, 2025. [arXiv:2506.01896](https://arxiv.org/abs/2506.01896).
- [G2026] Griego, Sebastian. Base-$21$ digit construction certificate for $C_{3a}$, [submitted to this repository](https://github.com/teorth/optimizationproblems/pull/71) (2026).
- [K2026] Kleinwaks, Logan. A masked-digit lower bound for the Gyarmati–Hennecart–Ruzsa sum–difference constant, [proof and verification package](https://github.com/kleinwaks/masked-digit-sum-difference-bound), [submitted to this repository](https://github.com/teorth/optimizationproblems/pull/134) (2026).