Timeline for Understanding sphere packing in higher dimensions
Current License: CC BY-SA 3.0
16 events
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| Jul 31, 2019 at 16:11 | comment | added | user929304 | @HenryCohn Hi, I was wondering if hopefully time allows, if you could have a look at this recent post, your input would be very valuable. Many thanks in advance | |
| Apr 12, 2016 at 21:57 | comment | added | Henry Cohn | Nope, your intuitions seems good (they just don't account for exceptional cases such as the Leech lattice). | |
| Apr 12, 2016 at 21:23 | comment | added | Henry Cohn | There's probably a counterexample with $n=23$, since the best sphere packing known in 23 dimensions is a little more than 1% less dense than the Leech lattice. However, we don't know whether it is optimal. This inequality presumably holds almost always (I don't know of another counterexample offhand). | |
| Apr 12, 2016 at 21:17 | comment | added | user88381 | I was wondering, is it in general the case that $\Delta_{n+1} \le \Delta_n$? My thought is that yes, because we know that in higher dimensions, volume is concentrated at the boundaries, thus interactions become more important, i.e. it becomes increasingly more difficult to pack as densely. Another intuitive reason (but possibly very wrong) would be that free spaces between spheres grow with dimension (since even for very small epsilon, $vol(B^n(1-\epsilon))/vol(B^n(1)) \to 0$ with n large. Eager to hear your take on this. | |
| Apr 10, 2016 at 20:16 | comment | added | Henry Cohn | Yes, it's essentially the Fourier transform of that inequality. | |
| Apr 9, 2016 at 14:12 | comment | added | Henry Cohn | Yup, this sounds like a reasonable outline. You are right that it's like Lagrange multipliers (linear programming duality is a special case, for linear functions). | |
| Apr 8, 2016 at 16:56 | comment | added | user88381 | (...) knowing the sharp bound is reached for $n=8$ among others, the main breakthrough of M. Viazovska entailed finding the auxiliary function $f$ that gives the sharp bound. Have I gone terribly wrong anywhere in my rough sketch of the ideas behind? Moreover if I may, I have a question of understanding: so in spirit, going from the constraints on page 37, Delsarte's included, taking the linear combination of the constraints then optimizing is similar to using Lagrange multiplier methods for usual functions? Thanks very much in advance for your reply. Sorry if this became terribly long. | |
| Apr 8, 2016 at 16:55 | comment | added | user88381 | (...) we extend Thm 4.1. to obtain bounds for the spherical packing problem in Euclidean space. This extension entails substituting ultraspherical polynomials by exponentials, thus the corresponding linear combination of Delsarte inequalities, is basically the Fourier transform of an auxiliary function $f$ which we are looking for. Then applying linear programming bounds, we come to Thm 3.1. (Cohn and Elkies), which gives an upper bound for the sphere packing density in terms of the $f$ and $\hat{f},$ proven in the notes for periodic packing. Given this Thm of upper bounds, and (...) | |
| Apr 8, 2016 at 16:54 | comment | added | user88381 | (...cont.) Thm 2.1., which gives a lower bound on the total energy for any N-point code $C.$ Then we move to the spherical code problem, which counts the number of spherical caps of radius $\theta/2$ we can place on the surface of a sphere. Applying linear programming bounds of Thm 2.1. to this problem, we arrive at Thm 4.1. To do so we set the energy potential $f$ to zero as there are no interactions here. This Thm gives an upper bound for $|\mathcal C|$, i.e. the total number of points of a code with minimal angle $\theta.$ Finally, from the spherical code problem, (...) | |
| Apr 8, 2016 at 16:53 | comment | added | user88381 | @HenryCohn The lecture notes are incredibly well written, thanks so much for this. Here's my attempt at a rough summary in 3 comments: I see that we start first from distribution of points on the surface of an $n$ dim sphere. The lectures first deal with the energy minimization case, where there's interaction between the points, given by the potential function $f.$ The question becomes then, which code (config of points) has the distribution $A$ (fulfilling a set of constraints) that minimizes the total energy function. Then using the Delsarte inequalities, one arrives at (...) | |
| Apr 7, 2016 at 10:54 | comment | added | j.c. | The functions $f(r),\hat{f}(r)$ that go into the bounds must have roots (with certain multiplicities) whenever $r$ is equal to the length of a vector in the lattice. The functions constructed by Viazovska for $n=8$ (and their analogues in $n=24$) neatly address this requirement with a sine squared factor (compare Table 1 of the Cohn and Miller paper in Henry Cohn's answer to Eqs. 32 and 48 in Viazovska's paper as well as Eq. 2.5 and the equation after 3.3 in the $n=24$ paper). Stephen Miller points this out in his lecture; I didn't discover any of this. | |
| Apr 6, 2016 at 21:59 | comment | added | j.c. | I guess I can answer my own question: after watching Miller's lecture at IAS here youtube.com/watch?v=8qlZjarkS_g it becomes more clear to me that the fact that the lengths of vectors in the $E_8$ and Leech lattices are square roots of integers leads to the abovementioned "sine squared factor". The lengths of vectors in the hexagonal lattice do not have this property, so some other kind of factor would be needed, if this approach were to work. | |
| Apr 6, 2016 at 15:44 | comment | added | j.c. | Does it seem likely that a proof of the dimension 2 case via the linear programming bounds will follow soon? In your paper with Miller, you make the comment "We will focus on n=8 and 24, not only because these cases are more interesting, but also because they appear to be more similar to each other than either is to the n=2 case." However it was not clear what properties that was referring to. | |
| Apr 6, 2016 at 9:07 | vote | accept | CommunityBot | ||
| Apr 5, 2016 at 15:01 | comment | added | Henry Cohn | Lecture 5 in the notes explains how these functions are relevant (you could actually skip Lectures 2-4, which are useful background but not really needed for this purpose). The short answer is that the functions and their Fourier transforms have opposite signs at most places, which means you get a lot of information if you plug them into Poisson summation and compare the two sides of the identity. Explaining in detail takes some space, but it's all there in Theorem 3.1 in Lecture 5. | |
| Apr 5, 2016 at 14:39 | history | answered | Henry Cohn | CC BY-SA 3.0 |