From 60979ad7a8966ffc0751aedf1f6f9097e20ad48b Mon Sep 17 00:00:00 2001 From: QZP Date: Sun, 6 Sep 2026 13:32:41 +0800 Subject: [PATCH] Clarify superseded certificate and order 47a bounds --- constants/45a.md | 4 +++- constants/47a.md | 2 +- 2 files changed, 4 insertions(+), 2 deletions(-) diff --git a/constants/45a.md b/constants/45a.md index f9f73165..052f2b2a 100644 --- a/constants/45a.md +++ b/constants/45a.md @@ -27,7 +27,9 @@ $C_{45}$ is the asymptotic density (if it exists) of the set of odd integers tha | $0.107648$ | [CE2018] | | -## Certificate for the $0.490249407811155$ upper-density bound +## Certificate for the earlier $0.490249407811155$ upper-density bound + +The current recorded upper bound is $0.490180063290061$ (see the table above); this section preserves the earlier finite certificate for reference. Let $$ diff --git a/constants/47a.md b/constants/47a.md index abf49d83..26a73861 100644 --- a/constants/47a.md +++ b/constants/47a.md @@ -39,8 +39,8 @@ the optimal weak-type $(1,1)$ constant of the centered Hardy–Littlewood maxima | Bound | Reference | Comments | | ----- | --------- | -------- | -| $\dfrac{3}{4}-\dfrac{\sqrt{2}}{4}+\dfrac{\sqrt{6}}{2}\approx 1.6211915$ | [Ald2000] | Aldaz's Proposition 1.4 gives a lower bound in every dimension $n\ge 2$. Specializing the formula to $n=2$ gives the displayed value. [Ald2000-prop1.4] | | $\dfrac{11+\sqrt{61}}{12}\approx 1.5675208$ | [Mel2003], [Ald2011] | Melas proved $c_1=\dfrac{11+\sqrt{61}}{12}$. Since $c_{d+1}\ge c_d$, we get $c_2\ge c_1$. [Mel2003-c1-formula] [Ald2011-monotone] | +| $\dfrac{3}{4}-\dfrac{\sqrt{2}}{4}+\dfrac{\sqrt{6}}{2}\approx 1.6211915$ | [Ald2000] | Aldaz's Proposition 1.4 gives a lower bound in every dimension $n\ge 2$. Specializing the formula to $n=2$ gives the displayed value. [Ald2000-prop1.4] | ## Additional comments and links