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Clarify two presentation issues from #168 #172

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4 changes: 3 additions & 1 deletion constants/45a.md
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Expand Up @@ -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
$$
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2 changes: 1 addition & 1 deletion constants/47a.md
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Expand Up @@ -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] |
| $\dfrac{11+\sqrt{61}}{12}\approx 1.5675208$ | <a href="#Mel2003">[Mel2003]</a>, <a href="#Ald2011">[Ald2011]</a> | Melas proved $c_1=\dfrac{11+\sqrt{61}}{12}$. Since $c_{d+1}\ge c_d$, we get $c_2\ge c_1$. <a href="#Mel2003-c1-formula">[Mel2003-c1-formula]</a> <a href="#Ald2011-monotone">[Ald2011-monotone]</a> |
| $\dfrac{3}{4}-\dfrac{\sqrt{2}}{4}+\dfrac{\sqrt{6}}{2}\approx 1.6211915$ | [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

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