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Credit Beigel–Gasarch for the 205/12 square-difference construction #175

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Credit Beigel–Gasarch for the 205/12 square-difference construction #175

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1 change: 1 addition & 0 deletions README.md
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Expand Up @@ -158,6 +158,7 @@ Bounds for which the level of available verification is currently at minimal lev
- [15a](https://teorth.github.io/optimizationproblems/constants/15a.html) **improved upper bound:** $C_{15a} \leq 2.371177$ by [E. Dupont, M. Eisenberger, B. Kozlovskii, A. Mehrabian, F. J. R. Ruiz, A. See, R. Zhou, J. Alman, V. Vassilevska Williams, M. Balog](https://arxiv.org/abs/2608.16884), 17 Aug 2026.
- [43](https://teorth.github.io/optimizationproblems/constants/43a.html) **improved lower bound (unverified):** $C_{43} \geq 0.860*$ (exact $43/50$; certificate-layer result conditional on the lemma set of [KHSHGW2026](https://arxiv.org/abs/2601.22365)) by [J. Savva](https://doi.org/10.5281/zenodo.22223485), 1 Sep 2026.
- [88a](https://teorth.github.io/optimizationproblems/constants/88a.html) **improved upper bound:** $C_{88a} \leq 186$ via $\mathrm{DHL}[40,2]$, by [OpenAI](https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf), 30 Aug 2026, with a Lean 4 formalization conditional on three declared axioms.
- [4b](https://teorth.github.io/optimizationproblems/constants/4b.html) **attribution correction:** the $205/12$ lower bound $C_{4b}\ge 0.733412$ is due to [R. Beigel and W. Gasarch](https://arxiv.org/abs/0804.4892), 2008; the same exponent was later published independently by [M. Lewko](https://doi.org/10.37236/4656), 2015.

## Maintainers

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4 changes: 2 additions & 2 deletions constants/4b.md
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Expand Up @@ -16,7 +16,7 @@ Let $r(N)$ be the maximum size of a subset $A\subset\\{1,\dots,N\\}$ with no non
| ----- | --------- | -------- |
| $\tfrac12$ | Trivial / folklore (see [BG2008]) | Can use an arithmetic progression of spacing $p \asymp \sqrt{N}$ |
| $\frac12\!\left(1+\frac{\log 7}{\log 65}\right)\approx 0.733077$ | [Ruz1984] | Base-expansion construction |
| $\frac12\!\left(1+\frac{\log 12}{\log 205}\right)\approx 0.733412$ | [Lew2015] | Improves modulus and residue set in base expansion |
| $\frac12\!\left(1+\frac{\log 12}{\log 205}\right)\approx 0.733412$ | [BG2008] | Base-expansion with modulus $205$ and residue set $S=\{0,2,8,14,77,79,85,96,103,109,111,181\}$ (Theorem 3.7). The same exponent was published independently in [Lew2015] |

## Additional comments and links

Expand All @@ -41,4 +41,4 @@ Let $r(N)$ be the maximum size of a subset $A\subset\\{1,\dots,N\\}$ with no non

## Contribution notes

ChatGPT 5.2 Pro was used to prepare an initial version of this page.
ChatGPT 5.2 Pro was used to prepare an initial version of this page. The $205/12$ row was reattributed after checking arXiv:0804.4892 Theorem 3.7 against the Lewko 2015 EJC abstract, which records the same exponent.