Thanks. I guess what you've shown is that if two vectors a and b are at an angle such that cos(angle) > 1/2, then (a-b) is shorter than a. Don't we also need to say something about the angle between (a-b) and b? It's late at night and I can't figure out what that is. :-) I imagine we'd have to say that it's no further from 60° than the angle between a and b was (but actually that's not true, so I guess the claim is something else?)
Perhaps it would be easiest to understand with the example: we want to say that, though (11, 5) and (1, 10) make an angle very close to 60° (specifically, about 59.845°), there must exist a pair of vectors that make an angle even closer to 60°, and with smaller coordinates. What is this pair of vectors, and how do we arrive at it? (If one of them is their difference namely (10, -5), what is the other one?)
Also, I think the top-level comment here (by joefkelley) considered only vectors with positive coordinates (maybe?), so we may need to take that into account and/or explain why the coordinates are always positive.
Regarding the angle between (a-b) and b: you can repeat the process until you end up with two vectors that cannot be reduced with respect to one another. Eventually the angle will be between 60 and 120 degrees (because you can always negate a vector which turns an angle of x degrees into 180-x degrees). So this process of basis reduction does not make the angle closer to 60 necessarily, but merely puts the angle somewhere within the [60,120] range
When approximating 60 degrees from below, you will always be able to 'reduce' the length of the vectors (which likely reduces the individual coordinates due to the relation of the infinity norm and the Euclidian norm). This means you will need a different approach to the brute-force than taken in the top-level comment.
Only positive coordinates means that you are restricted to one of the quadrants. I suspect (but do not have time to figure out/prove right now, and it might not be true) that when your lattice contains vectors with optimal angle close to 60 degrees, that you can either rotate or otherwise find an isomorphic lattice that has only positive coordinates in its basis vectors. However, it may just be a side-effect of how the original list was produced.
You're still not addressing the main question: how do you prove that the closest approximation, among vectors of a given magnitude — whether you take this to be the infinity norm (max coordinate), the Euclidean norm, total length, whatever — will never approximate 60 degrees from below?
Yes, as you've said, when the angle between two vectors is less than 60 degrees, you can always “reduce” the length of the vectors, but what good is this reduction, if the angle actually gets farther away from 60 degrees? All it does is preserve the lattice, but that's not our goal here.
Thanks for explaining lattice basis reduction to me. Now I know more about it, which is great. I just don't see how it has any bearing on the question asked here (https://news.ycombinator.com/item?id=18538456): why is it that, when we enumerate the “best” solutions among those under a given magnitude, they always (after the first two) approximate 60° from above? Is it coincidence and does the pattern break down (will we see 59.999...x° eventually as an angle that cannot be beaten by smaller vectors), or is there a proof?
And if there is a proof, can you please illustrate with the example of (11, 5) and (1, 10), how to (procedurally) get a pair of vectors with smaller magnitude, but angle at least as close to 60° as in this pair? (Note that the lattice reduction step you gave takes the angle further away from 60°, so as far as I can tell it's not helpful.)
Perhaps it would be easiest to understand with the example: we want to say that, though (11, 5) and (1, 10) make an angle very close to 60° (specifically, about 59.845°), there must exist a pair of vectors that make an angle even closer to 60°, and with smaller coordinates. What is this pair of vectors, and how do we arrive at it? (If one of them is their difference namely (10, -5), what is the other one?)
Also, I think the top-level comment here (by joefkelley) considered only vectors with positive coordinates (maybe?), so we may need to take that into account and/or explain why the coordinates are always positive.