Timeline for Extremal problem for 2-dimensional lattices
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 26, 2023 at 14:50 | vote | accept | Mikhail Katz | ||
| S Jun 26, 2023 at 14:50 | history | bounty ended | Mikhail Katz | ||
| S Jun 26, 2023 at 14:50 | history | notice removed | Mikhail Katz | ||
| Jun 24, 2023 at 15:27 | answer | added | David E Speyer | timeline score: 3 | |
| Jun 24, 2023 at 2:37 | comment | added | David E Speyer | I can achieve $1.5$. Identify $L$ and $L^{\ast}$ with $\mathbb{Z}^2$ and let $B$ have unit ball the parallelogram with vertices $\pm (1.5,0)$ and $\pm (0.75,1.5)$. Note that the lattice points inside this parallelogram are $(0,0)$, $(\pm 1,0)$, $(0,\pm 1)$ and $\pm (1,1)$ but that the lattice points in the strict interior are only $(0,0)$, $(\pm 1,0)$, showing that $\lambda_2(B)=1$. I believe that $\lambda_1(B^{\ast}) = 1.5$. Furthermore, I think I can prove this is optimal. I'll try to write up my thoughts over the weekend. | |
| Jun 22, 2023 at 14:41 | comment | added | David E Speyer | I can improve the lower bound from $\sqrt{\tfrac{4}{3}} \approx 1.155$ to $\tfrac{4}{3} \approx 1.333$. Identify $B$ and $B^{\ast}$ with $\mathbb{R}^2$ and identify $L$ and $L^{\ast}$ with $\mathbb{Z}^2$. Let the norm on $B$ have unit ball the hexagon with vertices $\pm (1/2,1)$, $\pm (1, 1/2)$ and $\pm (1/2, -1/2)$. The unit ball in $B^{\ast}$ is the hexagon with vertices $\pm (2/3, 2/3)$, $\pm (-2/3,4/3)$ and $\pm (4/3,-2/3)$. I get that $\lambda_1(L) = \tfrac{4}{3}$ and $\lambda_2(L^{\ast}) = 1$. | |
| S Jun 22, 2023 at 9:13 | history | bounty started | Mikhail Katz | ||
| S Jun 22, 2023 at 9:13 | history | notice added | Mikhail Katz | Draw attention | |
| Jun 20, 2023 at 12:48 | history | edited | YCor | CC BY-SA 4.0 |
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| Jun 20, 2023 at 9:26 | history | edited | Mikhail Katz |
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| Jun 20, 2023 at 9:13 | history | asked | Mikhail Katz | CC BY-SA 4.0 |