# Zhenghe's Blog

## February 25, 2010

### Why golden mean is optimal: an introduction to Diophantine, Brjuno and Liouville number

Irrational rotations on unit circle $S^1=R/Z$ is one of most important examples in dynamical systems, which I am going to denote it by: $\alpha: S^1\rightarrow S^1, x\mapsto x+\alpha$, where $\alpha\in [0, 1]\setminus Q$.

Then we have the following facts:

Theorem 1: Dynamical systems defined above are minimal and uniquely ergodic.
(Th0se who are not familiar with these concepts can google them or take a look at Peter Walters’ “An introduction to ergodic theory”.)

The subtlety of these dynamical systems lie in the way the orbit Orb(x) of a point x recurrent to itself, in particular, we can fix x=0. It turns out the recurrence of 0 is closely related to the arithmetic property of the rotation angle $\alpha$, which lead to interesting classification of irrational numbers. WLOG, I am going to restrict myself to I=[0,1].

Definition 2: $\alpha$ satisfies a Diophantine condition $DC(\gamma, \tau), \gamma>0, \tau>0$, if $|q\alpha-p|>\gamma |q|^{-\tau}, (p, q)\in Z^2, q\neq 0$.

Let $DC=\cup_{\gamma>0, \tau>0}DC(\gamma, \tau)$ be the so-called Diophantine number, then it is well known that $DC_{\tau}=\cup_{\gamma>0}DC(\gamma, \tau)$ has full Lebesgue measure if $\tau>1$. In some sense, Diophantine condition means that $\alpha$ cannot be approximated by rational numbers too fast, which in dynamics language means 0 cannot be recurrent to itself too fast under rotation $\alpha$

There is actually another equivalent way to define Diophantine number in terms of continued fraction expansion of $\alpha$.
Let $\alpha=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots}}}}$ be the continued fraction expansion of $\alpha$, and $\frac{p_n}{q_n}=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{\ddots+\frac{1}{a_n}}}}}$ be the n’th step approximant, then we have the following properties:

(1)$\frac{1}{q_n}>|q_{n-1}\alpha-p_{n-1}|>\frac{1}{q_n+q_{n-1}}$,
(2)$\parallel k\alpha\parallel_{R/Z}>|q_{n-1}\alpha-p_{n-1}|, \forall q_{n-1}+1\leq k\leq q_n-1$.
where $\parallel\cdot\parallel_{R/Z}$ denote the distance to the nearest integer.

Thus, to define Diophantine condition, we need only care about the continued fraction approximants, where then Diophantine conditions are actually given by the growth rate of sequence $(q_n)_{n\in Z^+}$. Use above inequalities and definition 2, it’s not difficult to see that:

$\alpha\in DC_{\tau}$ iff $q_{n+1}, where c is some constant.

But what about the growth rate of $(q_n)_{n\in Z^+}$ in general? In fact, we have the following nice induction formula for $(q_n)_{n\in Z}$:

$q_n=a_nq_{n-1}+q_{n-2}$.

Thus, the slowest growth is when $a_n=1$ for all n, where the corresponding $\alpha$ is the golden mean and $(q_n)_{n\in Z}$ is the Fibonacci sequence if we set $q_0=1, q_{-1}=0$! So in some sense, golden mean is the most irrational irrational number, because it is approximated by rational numbers in the slowest rate.

For Fibonacci sequence $b_n$, we can find the exact expression by generating function, which is

$b_n=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^{n+1}-(\frac{1-\sqrt{5}}{2})^{n+1})$.

Hence, sequence $(q_n)_{n\in Z^+}$ grows at least exponentially fast.

Definition 3: $\alpha$ is called a Brjuno number if the associated sequence $(q_n)_{n\in Z^+}$ satisfies $\sum_n \frac{log q_{n+1}}{q_n}<\infty$.

By the equivalent condition of Diophantine number and the slowest growth of $(q_n)_{n\in Z^+}$, Brjuno number contains all Diophantine number so it’s a full Lebesgue measure set. We can define another number which decribe the growth rate of $(q_n)_{n\in Z^+}$ outside a set contains Brjuno numbers.

$\beta(\alpha)=\limsup\limits_{n\rightarrow\infty}\frac{log q_{n+1}}{q_n}$.

Then obviously we have:

$DC\subset$ Brjuno Numbers $\subset\{\alpha: \beta(\alpha)=0\}$.

Again $\{\alpha: \beta(\alpha)=0\}$ is a full Lebesgue measure set. Thus,  it seems that $\{\alpha: \beta(\alpha)>0\}$ is a rather small set which may even be empty. But that’s not true:

Definition 4: $\alpha\in${$\alpha: \beta(\alpha)=\infty$} are said to be Liouville number.
Remark: Some also define Liouville numbers to be the complement of Diophantine numbers in irrational numbers.

Obviously, the set of all Liouville numbers has zero Lebesgue measure and they are approximated by rational numbers really very fast. It is an interesting fact that the set of Liouville numbers is generic though it has zero Lebesgue measure. These arithmetic properties in many ways govern dynamical behaviors in many dynamical systems.