Timeline for Extremal problem for 2-dimensional lattices
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 6, 2023 at 21:15 | comment | added | Tom Goodwillie | How do you rule out the possibility of there being no non-zero lattice points at all in the interior of $K$? I think an additional argument is needed. For example by perturbing the norm slightly without introducing more lattice points to $K$, you can arrange for one of the boundary lattice points to be now in the interior, in such a way that the new norm never increases and therefore the new dual norm never decreases and therefore $\lambda_1$ does not decrease. | |
| Jun 27, 2023 at 13:58 | comment | added | Mikhail Katz | I sent you Averkov's explanation via email. | |
| Jun 27, 2023 at 13:50 | comment | added | David E Speyer | Okay, I guess I'm wrong, I'm not claiming to have a proof for the $\lambda_1(L) \lambda_1(L^{\ast})$ optimum. I'd be curious to see the example that achieves $\sqrt{2}$. | |
| Jun 27, 2023 at 13:39 | comment | added | Mikhail Katz | Wait a minute, but Mahler seems to say otherwise: $\lambda_1(L)\lambda_1(L^\ast)$ reaches $\sqrt2$. A colleague sent me an explanation of how this follows from Mahler's '48 paper. | |
| Jun 27, 2023 at 13:35 | comment | added | David E Speyer | For this particular lattice, $\lambda_1(L) = 2/3$ (because shrinking $P$ by $2/3$ still includes $\pm (1,0)$, so $\lambda_1(L) \lambda_1(L^{\ast}) = 1$. I suspect that the optimal value of $\lambda_1(L) \lambda_1(L^{\ast})$ is $4/3$, using the hexagon I posted as a comment earlier. | |
| Jun 27, 2023 at 8:31 | comment | added | Mikhail Katz | Awsome! Here your convention is to work with $\lambda_2(L)\lambda_1(L^\ast)$. How big is the difference between $\lambda_1(L)$ and $\lambda_2(L)$ for this extremal lattice? | |
| Jun 26, 2023 at 14:50 | vote | accept | Mikhail Katz | ||
| Jun 26, 2023 at 14:50 | history | bounty ended | Mikhail Katz | ||
| Jun 25, 2023 at 8:38 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
typo
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| Jun 24, 2023 at 16:18 | history | edited | David E Speyer | CC BY-SA 4.0 |
deleted 7 characters in body
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| Jun 24, 2023 at 15:27 | history | answered | David E Speyer | CC BY-SA 4.0 |