Timeline for maximizing number of lattice points with bounded diameter
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Feb 14, 2023 at 19:59 | comment | added | Mark Schultz-Wu | it's worth mentioning there are more refined arguments available, at least in the setting of $\ell_2$. For a lattice $L$ one can often get estimates for $|L\cap (B_2^n(r) +t)|$ in terms of the gaussian mass of the lattice. See the introduction of noah stephens-davidowitz's thesis for some exposition on this. I don't know if the gaussian arguments extend past $\ell_2$ though. | |
| Feb 14, 2023 at 19:52 | comment | added | Mark Schultz-Wu | bound, but don't recall it offhand. | |
| Feb 14, 2023 at 19:52 | comment | added | Mark Schultz-Wu | it isn't rigorous in that it leads to an equality, but you can get bounds. For example, it is straightforward to see that $|S\cap\mathbb{Z}^n| \leq \mathsf{vol}(S + \mathcal{B}_\infty^n(1/2))$, where $+$ is the Minkowski sum, and $B_\infty^n(1/2) = [-1/2,1/2]^n$ is the $\ell_\infty$ ball of radius $1/2$. Then, if you care about $\ell_1$ diameter, you can write something like $S\subseteq B_1^n(\mathsf{diam}(S)/2)$ and $B_\infty^n(1/2) \subseteq B_1^n(n/2)$ to get an upper bound on the point count in terms of solely $\ell_1$ quantities. I remember there being a similar argument for the lower | |
| Feb 14, 2023 at 19:18 | comment | added | Yan X Zhang | Yeah I feel the hard part is making the "point count to volume" rigorous here, but this is a novel direction for me to search / learn as well, so thanks! | |
| Feb 14, 2023 at 10:00 | comment | added | Mark Schultz-Wu | count, rather than a volume, and is $\ell_1$, and not $\ell_2$, but hopefully this is still somewhat useful. | |
| Feb 14, 2023 at 10:00 | comment | added | Mark Schultz-Wu | You want to maximize $|S|$ given some diameter constraint $d$. It seems equivalent to maximize $\frac{|S|}{d^n}$, where $d^n$ is introduced to make the quantity scale-invariant (sort of --- here I am being imprecise and identifying $|S|$ with volume of the convex hull of $S$). Equivalently, you can minimize $\frac{d^n}{|S|}$. At least in $\ell_2$, this is often called the "thickness of the lattice covering", and there is quite a bit written about it, say for example in Conway and Sloane's Sphere Packings, Lattices, and Groups. This isn't an answer as your setting both has $|S|$ as a point | |
| Feb 14, 2023 at 2:25 | history | asked | Yan X Zhang | CC BY-SA 4.0 |