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Apr
12
comment Understanding sphere packing in higher dimensions
Nope, your intuitions seems good (they just don't account for exceptional cases such as the Leech lattice).
Apr
12
comment Understanding sphere packing in higher dimensions
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
10
comment Understanding sphere packing in higher dimensions
Yes, it's essentially the Fourier transform of that inequality.
Apr
9
comment Understanding sphere packing in higher dimensions
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
6
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5
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Apr
5
comment Understanding sphere packing in higher dimensions
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
answered Understanding sphere packing in higher dimensions
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Mar
14
answered Could this unexpected bias in the distribution of consecutive primes have any impact on the security of encryption algorithms?
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Sep
17
comment electron configuration on manifolds
Nope, it's generally far from unique, even on spheres. The first case seems to be 16 points on $S^2$, for which there are (at least) two local optima. As the number of points grows, the number of local minima seems to increase exponentially, but no proof is known. In higher dimensions there are cases with arbitrarily large numbers of non-isometric global minima (the configurations in the last line of Table 1 in arxiv.org/abs/math/0607446).
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