What's the Geometry of Numbers? - Minkowski's Theorem #SoME2

We're looking at Minkowski's Geometry of Numbers Theorem and applying it to prove the so-called Fermat's Christmas Theorem. #SoME2
Timetable:
0:00 - Introduction
1:55 - Symmetric Convex Bodies
3:28 - Proving the Main Theorem
7:00 - Other Lattices
7:44 - Fermat's Christmas Theorem
10:35 - Other Questions

[includecmath]

I'm currently writing up some lecture notes where I use Minkowski's Theorem to prove Lagranges 4 Square Theorem, so I was particularly interested when I saw your video. I really like your presentation. You are rigorous and the visuals are pretty nice. I also like that you not just show the Theorem but also apply it to a well known problem. Great job!

[AaronRotenberg]

A very nice topic that I've not heard of before.

In the proof at 3:24, you don't actually state why the set needs to be bounded and open. I thought about it, and actually this condition is just ensuring the set is measurable. So a more rigorous version of this proof would be:

Let M be a measurable subset of R² with measure greater than 1. Consider the countable collection M' of measurable sets obtained by intersecting M with the integer square lattice; we have that the sum of the measures of each set in M' equals the measure of M. Now translate each set in M' by the appropriate integer vector offset to the origin. Then we have a countable collection of measurable sets, so they have a measurable union; and that union is a subset of the unit square, so its measure is not greater than 1. Now suppose the translated M' are pairwise disjoint. Then their union's measure would equal the sum of their measures. But this contradicts that the measures of M' sum to a value greater than 1, so the assumption of disjointness must be false.

Wikipedia says there is a stronger version of these theorems called Blichfeldt's theorem that actually gives a lower bound on the number of lattice points that have to be contained within a set with a given area.

[drdca]

This is nice! I wasn’t familiar with either the lemma, nor either of the theorems.

Nitpick(s) below the readmore line


At one point you accidentally say that x^2 + y^2 “is the square root of the distance from the origin”, when you meant either that the square root of x^2 + y^2 is the distance, or that x^2 + y^2 is the square of the distance.

[bygradforgrad]

Excellent video. I did my PhD dissertation dealing with a generalization of this concept.

[johnchessant]

Great video, really cool proof of the sum of squares theorem!

[milenamarquez]

Please please make a new video on this topic! Looking forward to the sequel :)

[amankarunakaran]

Fantastic video. This was sweet - I only have a couple criticisms. One is pacing - I think somewhat obvious stuff was presented at the same pace as less obvious stuff; the claim at 4:04 was really not obvious enough imo to be stated at a conversational pace and then move along to the next thing, it is worth at least drawing an example for why it's true. Also, at 4:30 - why should I imagine it?? You have the visualization tool, not me! :) that was a great opportunity to show the squares overlap with one another and highlight the intersection, and make a continuous analogue to the pigeonhole principle.

The primary other criticism I have is not unique to you; it's true of most math presentations. In fact, it may not necessarily even be applicable depending on what your intent of the video is. If this is meant to be a cool visual reference of the proof of an interesting result, then I think it has accomplished that job well. If, on the other hand, this is meant to be an instructional video/tool, I think the criticism applies.

The common presentation of mathematical information in textbooks, classes, and most parts of this video goes something like that the theorem answers, but most mathematical discovery often works like examples->questions about those examples->conjecture->proof. When presented in the second manner, the motivation becomes a lot more clear and the viewers can often be much more engaged (because now they are participants - they have asked the same questions about the examples, and they want to know the answer!). The video starts out strong in this regard, and draws various shapes on the plane, asking questions about the non-origin integer lattice points contained in the shape. But then, the restriction to symmetric convex bodies comes out of nowhere! Imo it makes more sense to "play the game" of how big can I make my shape without containing integer points, but then the viewer will probably arrive at the conclusion that non-convex shapes can get arbitrarily large, at which point it then makes a lot more sense to introduce the convexity restriction. Similar "playing around with examples" demonstrations can be done with the other conditions. At 3:30, it is so not obvious to me as a viewer why I would want to prove this in order to answer the larger theorem. Imo it is not hard to conjecture that the area constant in Minkowski's theorem is 4 if you play around with examples, trying to make your shape as large as possible without containing lattice points. At that point, trying to use the properties of convexity to force a lattice point to exist can guide you to the idea of a difference of two points being a lattice point, which then motivates the lemma at hand.

The part of the video after Minkowski's theorem is awesome but suffers from a similar issue to the one I mentioned above, a lot of information is thrown at you without a lot of motivation as to how one could come up with that, and so it's fine if you pause the video and grok it yourself, but imo making a separate video with more drawn out motivation would be clearer as an instructional tool.

All in all, only my perspective so I could definitely be wrong about some of the stuff I've written, but this was great!

[itellyouforfree]

Simple solution to 11:10. Dirichlet theorem: for every real number c there are infinite rational approximations m/n that satisfy |c-m/n| < 1/n^2. Let the line be y=cx. Consider the point (n, m). Then distance(point, line) <= |m-cn| < 1/n, so we can find points with arbitrary small distance.

[juanramonvazquez]

Pretty nice video, even more so for a first video. Animations and scripts are great (excluding that mishap at 8:20 when explaining the pythagorean formula)

[ihp]

Thank you, please post more videos like this!

[rupen]

This video goes really well with the other #some2 video "Why do we care about functions?" by Chillaxiom.

[Number_Cruncher]

Very nice. Can't wait to see the follow ups.

[itellyouforfree]

8:55 instead of u = -1 (mod p) it should be u^2 = -1 (mod p)

[pra.]

Very cool! Like the ending problems :)

[joseville]

10:00 I may have missed something, but why does p divided a^2 + b^2?

Also, where did the theorem "there exists u: p | u^2 + 1..." come from? Is it true for any prime p? Or only the ones of the form 4k+1?

[dyld]

Just one feedback: Your explanation of the main theorem at 4:04 was very confusing and took me a while to understand. It's partly because the animation doesn't go along with your words, so I didn't fully understand what you meant. For example, when you say "there are two points with the same position relative to the squares", actually show those points! And when you say "imagine taking the square tiles and putting them on top of each other", show it in the animation! Not everyone has a visual mind.

[joseville]

Awesome video!

Does symmetric in this case mean 180 deg rotation about the origin symmetry?

Note, that after 180 deg rotation about the origin, the point (x, y) becomes (-x, -y)

[fullfungo]

At 8:55 did you mean u^2 = -1 (mod p) instead of u = -1?

[alitalalhaidar]

Every time you open a box, and you insert a lemma or a theorem in it, you may want to add a figure(s) in the box

[columbusmyhw]

At 8:55 you mean u^2=-1(mod p), not u=-1(mod p)