From 1360842a99f302b41ee12b555379c6582c8ada56 Mon Sep 17 00:00:00 2001 From: Logan Kleinwaks Date: Fri, 24 Jul 2026 15:06:18 -0400 Subject: [PATCH 1/5] Improve lower bound for C_3a to 1.19023813 --- constants/3a.md | 13 ++++++++----- 1 file changed, 8 insertions(+), 5 deletions(-) diff --git a/constants/3a.md b/constants/3a.md index 7c84b23..8bb0404 100644 --- a/constants/3a.md +++ b/constants/3a.md @@ -1,4 +1,4 @@ -# The Gyamarti-Hennecart-Ruzsa sum-difference constant +# The Gyarmati-Hennecart-Ruzsa sum-difference constant ## Description of constant @@ -18,18 +18,20 @@ $$ |A-B| \gg |A+B|^{C_{3a}}.$$ | 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. | +| $1.19023813$ | [K2026] | Base-$6169$ masked-digit limit construction with $M=\langle312,315,336,416,420\rangle\cap[0,3084]$ 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 $1.25$. +- [K2026] generalizes the bounded-digit limit construction in [Z2025] by replacing bounded digits with digits restricted to a mask. - [AlphaEvolve repository page for this problem](https://google-deepmind.github.io/alphaevolve_repository_of_problems/problems/44.html) ## References @@ -40,3 +42,4 @@ $C_{3a} \geq 1 + \log( \lvert U-U \rvert /\lvert U+U \rvert )/\log(2 \max(U)+1)$ - [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/NNN) (2026). From ef5de2b566fd13ff027d76115de2abf75f4de8c4 Mon Sep 17 00:00:00 2001 From: Logan Kleinwaks Date: Fri, 24 Jul 2026 15:10:05 -0400 Subject: [PATCH 2/5] Update comment about mask in [K2026] generalization of [Z2025] --- constants/3a.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/constants/3a.md b/constants/3a.md index 8bb0404..9f06321 100644 --- a/constants/3a.md +++ b/constants/3a.md @@ -31,7 +31,7 @@ $$ |A-B| \gg |A+B|^{C_{3a}}.$$ - 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 $1.25$. -- [K2026] generalizes the bounded-digit limit construction in [Z2025] by replacing bounded digits with digits restricted to a mask. +- [K2026] generalizes the bounded-digit limit construction in [Z2025] by replacing bounded digits with digits restricted to a finite mask. - [AlphaEvolve repository page for this problem](https://google-deepmind.github.io/alphaevolve_repository_of_problems/problems/44.html) ## References From 61a7129ee1455eb56a990973ad0b91af085b0b4c Mon Sep 17 00:00:00 2001 From: Logan Kleinwaks Date: Fri, 24 Jul 2026 16:10:16 -0400 Subject: [PATCH 3/5] Update [K2026] reference with pull request URL --- constants/3a.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/constants/3a.md b/constants/3a.md index 9f06321..0e43bf4 100644 --- a/constants/3a.md +++ b/constants/3a.md @@ -42,4 +42,4 @@ $C_{3a} \geq 1 + \log( \lvert U-U \rvert /\lvert U+U \rvert )/\log(2 \max(U)+1)$ - [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/NNN) (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). From 5c5f188bbcada37aed90f544b729a1729321411b Mon Sep 17 00:00:00 2001 From: Logan Kleinwaks Date: Sun, 26 Jul 2026 12:38:05 -0400 Subject: [PATCH 4/5] Update C_3a record to 1.19102809 --- constants/3a.md | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/constants/3a.md b/constants/3a.md index 0e43bf4..d2eddb9 100644 --- a/constants/3a.md +++ b/constants/3a.md @@ -24,14 +24,15 @@ $$ |A-B| \gg |A+B|^{C_{3a}}.$$ | $1.173077$ |[Z2025]| | | $1.1740744$ | [G2026] | Base-$21$ digit construction with exact counting certificate. | | $1.1835129324$ | [MI2026] | Base-$33$ digit construction with exact counting certificate. | -| $1.19023813$ | [K2026] | Base-$6169$ masked-digit limit construction with $M=\langle312,315,336,416,420\rangle\cap[0,3084]$ and a directed-rounding 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 From b114732fba7b45c221a198f107e99222c1b8f7fe Mon Sep 17 00:00:00 2001 From: Logan Kleinwaks Date: Sun, 26 Jul 2026 12:39:18 -0400 Subject: [PATCH 5/5] Fix markdown in 3a.md --- constants/3a.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/constants/3a.md b/constants/3a.md index d2eddb9..e307d7a 100644 --- a/constants/3a.md +++ b/constants/3a.md @@ -32,7 +32,7 @@ $$ |A-B| \gg |A+B|^{C_{3a}}.$$ - 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 $1.25$. - [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 $\langle506,508,529\rangle=23\langle22,23\rangle+508\mathbb N_0$. +- 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