From Diophantine approximation to geometry of numbers

We all know about Pythagoras and his obsession with triangles. Usually when we first learn about the rational and real numbers, we are also told about the Pythagoreans, a 6th century BCE cult that started with the followers of Pythagoras and which dealt with mysticism and Mathematics (and if it still exists and you are a member in it, please send me a message – I would love to join). The legend says that they loved ratios between natural numbers, i.e. rational numbers, and didn’t believe that there exist any other type of numbers. Unfortunately, Hippasus which was one of their own “betrayed” them and proved that \sqrt{2} is not rational. When he died by drowning, the Pythagoreans said that this was a punishment by the gods themselves for showing that the world of Mathematics is not perfectly rational. Well, it was either this, or because he built a dodecahedron inside a sphere. Anyway, these days we know that there are irrational numbers, of course, but it is nice to know that if you insult the gods of Mathematics you will be punished.

The moral from this story is that rational numbers are divine, but there are other types of numbers – like the real numbers. While there are irrational numbers, there are so many rational numbers so that we can get as close as we want to any real number. In this post we will try to investigate when should we say that an approximation is good, how good of an approximation can we hope to find for a given real number, and what is the connection to lattices (see the post about lattices for the definitions and some examples).

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A wild lattice appears

A couple of years ago I began doing some research in a new mathematical area (at least new for me) and suddenly lattices began to appear everywhere. The thing that makes these lattices interesting is how they are connected to other mathematical areas. These type of connections, which appear time and time again in mathematics is what makes it so beautiful. As I advance more and more in my studies, I decided that it is time to write some of it down for other people to learn and enjoy.

This series of posts will show how these lattices are connected to several mathematical areas that I find very interesting, with emphasize on number theory and Diophantine approximation. At least for the first posts in the series, I will keep the math as simple as possible so it would be accessible to any student with basic mathematical knowledge (some linear algebra and a little bit of group and ring theory). Hopefully, by the time I will get to write the more advanced material, you will be interested enough in order to fill in the details.

In this post we will see the most basic definitions and results about lattices which we will need for the rest of this series of posts.

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The generic matrix and the Cayley–Hamilton theorem

Sometime during the first course in linear algebra we all learn the famous Cayley-Hamilton theorem which states the following:

Theorem: Let A \in M_n (F) be an n \times n matrix over a field F, and denote by p_A(t)=\det(tI-A) its characteristic polynomial. Then p(A)=0.

The “easy” proof for this theorem is just noting that p_A(A)=\det(AI-A)=det(0)=0. Of course, this is not a real proof since in the definition as a determinant of p_A(x), the symbol x is a place holder for a scalar and not a matrix. Nevertheless, the theorem is still true and has several proofs. For example, if you believe in the Jordan presentations or that each matrix can be approximated by a diagonalizable matrix, then this theorem is not that far fetched, since it is true for matrices in the Jordan form and for diagonlizable matrices.

Our goal here is to show that the most “generic” matrix satisfies this theorem, so all the other matrices have no real choice, and must follow the footsteps of their generic master ruler and do the same. Continue reading

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