Diophantine approximations is a field in number theory which asks how good a real number can be approximated by some rational number. Quite a number of classical results are proved by using combinatorics, which is not friendly to people who cannot count. Here I will give a proof of some Diophantine approximations using only point set topology/dynamical system, or more precisely topological dynamics. (Main reference is this book.)
Throughout whole post, will be a compact topological space, and is a group or semigroup acting on by continuous transformations (i.e. for some continuous ). In this post, most of the time . In this case, is generated by , and we denote by the transformation on representing the action of the element . We then denote the dynamical system by and call it a cyclic system.
Definition 1 Let be a continuous map of a topological space into itself. A point is called a recurrent point for (or for ) if for any neighbourhood containing , there exists such that .
It means that if is a recurrent point, then it will eventually go back to somewhere near . Surprisingly, every continuous map on a compact space has at least one recurrent point:
Theorem 1 (Birkhoff’s recurrence theorem) If is a continuous map of a compact space to itself, then the set of recurrent points for in is non empty.
Proof. Let , ordered by inclusion. Let be a totally ordered chain. Then for all . The intersection is non empty because of compactness. By Zorn’s lemma, has a minimal element. Let be minimal. We claim that every point in is recurrent. Let , and let . Note that since is closed and invariant under . But then so is . By minimality, . Let be a neighbourhood of containing . If , then we are done. If not, then , and so . But , contradiction.
Although we used Zorn’s lemma, it is known that the above result is independent of Zorn’s lemma.
In some special cases, we can guarantee that there are more recurrent points.
Definition 2 Let be a compact group, let and let be defined by . We call the dynamical system a Kronecker system.
Theorem 2 Every point in the space of a Kronecker system is recurrent.
Remark: Kronecker system was not invented by Kronecker. Reason for this terminology is that one of his theorem is equivalent to Theorem 2, with is torus instead of general compact group.
Proof. By Theorem 1, there is a recurrent point . Let be arbitrary. Then . Let be a neighbourhood of . Then is a neighbourhood of . So there exists such that , i.e. .
Now we talk about how to construct a new dynamical system by using some given dynamical system(s).
Definition 3 Let and be two dynamical systems with the same semigroup of operators. A homomorphism from to is given be a continuous map satisfying
(Note that the products and refer to the action of on two different spaces.) A dynamical system is a factor of the dynamical system if there exists a homomorphism from onto . In this case we also say that is an extension of .
Phenomena taking place on generally carry over to factors of . For example:
Proposition 3 If determines a homomorphism of a cyclic system to and is recurrent for , then is recurrent for .
Proof. Let be recurrent. Let be a neighbourhood of . By continuity of , is a neighbourhood of . Since is recurrent, there exists such that . Hence , and so . Therefore is recurrent for .
We can also form some special product:
Definition 4 Let be a dynamical system, be a compact group and be continuous. Let . Define by Then is called a group extension of , or sometimes called a skew product of with .
Remark: If is a group extension, then the elements of act on by right translation, i.e. . It is easy to show that . As a result is an automorphism of the system .
The reason of forming such a weird product because we want to carry the good property of Kronecker system to the extended dynamical system:
Theorem 4 Let be recurrent for , and let be a group extension of , . Then each of the points , where , is a recurrent point of .
Proof. Let denote the identity of . We claim that is a recurrent point of . Let and let . Note that is recurrent if and only if . Now, since is recurrent for , . Notice that , and so . Therefore , where . Now , where the first equality comes from the remark above, we have . Using the fact that the relation “” is transitive, we obtain that , and thus by induction we have for all . By Theorem 2 and the fact that is closed, we have , i.e., is recurrent. By Proposition 3, is recurrent, i.e. is recurrent.
Some examples of non-Kronecker dynamical systems for which every point is recurrent:
- Let , and let be defined by , where is fixed.
Notice that is a group extension of the Kronecker system on the circle , with . By Theorem 2 and Theorem 4, every point of is recurrent. In particular, is recurrent. Now consider the orbit
As a by-product, since is recurrent, for any , we can solve the Diophantine inequality , or , becasue the second entry can be arbitrarily close to in , i.e. an integer.
- More generally, let be a polynomial of degree with real coefficients. Write and form successively , , and so on. Then is of degree . Let be the constant . Define by . This defines a group extension of a dynamical system on the -torus, which in turn is a group extension of a dynamical system om the -torus, and so on. Consider the orbit of the point . Since , we obtain that , and so . Therefore comes arbitrary close to modulo 1. Thus, again as a by-product, if is a real polynomial with , then for any , we can solve the Diophantine inequality .