Timeline for The range of each of successive minima for all unimodular lattices
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 20, 2023 at 13:14 | comment | added | Mikhail Katz | I posted this at mathoverflow.net/q/449248/28128 in case you wish to share any insight. | |
| Jun 19, 2023 at 18:32 | comment | added | Mark Schultz-Wu | I don't know of any works doing precisely what you want. It is possible you could use Kissing Numbers and Transference Theorems from Generalized Tail Bounds. They essentially tried optimizing (by reducing constant factors) + generalizing Banaszczyk's argument, and reduced the problem of general transference inequalities to finding a suitable function $f$ that satisfies certain properties, which might be easy in your setting. | |
| Jun 19, 2023 at 16:29 | comment | added | Mikhail Katz | P.S. I have a related text that provides motivation; if you are interested, give me an email and I'll send a pdf. | |
| Jun 19, 2023 at 16:28 | comment | added | Mikhail Katz | Yes, that gives you $\sqrt{\frac83}$ but I was wondering if anybody calculated the optimal bound in dimension 2. Banaszczyk's techniques give good asymptotic estimates but don't necessarily give the best results in low dimensions. | |
| Jun 19, 2023 at 16:23 | comment | added | Mark Schultz-Wu | Banaszczyk's techniques extend pretty directly to ellipsoids (see Inequalities for Convex Bodies and Polar Reciprocal Lattices in $\mathbb{R}^n$). In 2 dimensions any convex body (= norm ball) is well-approximated by an ellipsoid (c.f. John ellipsoids). Applying these techniques to the John ellipsoid of your norm ball should get you what you want. | |
| Jun 19, 2023 at 11:32 | comment | added | Mikhail Katz | Hi Mark, would you happen to know what is known about the 2-dimensional version of the problem when one works with arbitrary Banach norms rather than the Euclidean norms, such as the optimal value for $\lambda_1(L) \lambda_2(L^\ast)$ ? | |
| Mar 29, 2022 at 8:39 | history | answered | Mark Schultz-Wu | CC BY-SA 4.0 |