Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I solved an exercise in which the first part asks to prove that for any measure preserving measurable transformation $f:[0,1]\rightarrow [0,1]$ we have $$\liminf_{n} n |f^n(x)-x| \leq 1, \ \mbox{a.e.}$$

I can't prove the second part of the exercise: Let $\omega=(\sqrt{5}-1)/2$ and let $f:[0,1]\rightarrow[0,1]$ defined as $f(x)= (x+\omega) \pmod{1}$. Use this transformation to prove that there is no $c<\frac{1}{\sqrt{5}}$ such that $$\liminf_{n} n |f^n(x)-x| \leq c$$

Thank you guys in advance! (I'm sorry about the mistakes!)

share|cite|improve this question
How did you proof the first part? If you take $f=0$ and $x=\frac{1}{2}$ the bound is not satisfied. – Lucien Feb 12 '13 at 16:34
I'm sorry! I forgot to say that it holds a.e. – badaui90 Feb 12 '13 at 16:50
If you take $f=0$, then still the bound is satisfied only for $x=0$, so not a.e. – Lucien Feb 12 '13 at 17:00
My bad again, I was think about the problem and forgot to say that $f$ also has to preserve Lebesgue $\lambda$ measure, ie,$\lambda(f^{-1}(B))=\lambda(B)$ for all measurable set $B$. – badaui90 Feb 12 '13 at 17:13
up vote 1 down vote accepted

As long we are dealing with $\omega \pmod 1$ it doen't make any difference to deal with $\omega+1 = \frac{\sqrt{5}+1}{2}= \phi$. Therefore we can apply Hurwitz's theorem that states: for every irrational number $\zeta$ there are infinitely many rationals $m/n$ such that $$\left| \zeta - \frac{m}{n} \right| \leq \frac{1}{\sqrt{5} n^2}$$ Moreover $\sqrt{5}$ is the best constant you can get: if you replace it with an $A>\sqrt{5}$ and take $\zeta=\phi$ there are only a finite number os rational numbers suck that the propertry above holds with $A$ instead of $\sqrt{5}$.

Here you can download a book from Carlos Gustavo Tamn de A. Moreira (Gugu), a researcher at IMPA - Rio de Janeiro, Brazil, where at page 47 you can see this result. (proof starts at page 60)

The original paper is: Hurwitz, A. (1891). "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (On the approximation of irrational numbers by rational numbers)" (in German). Mathematische Annalen 39 (2): 279–284. doi:10.1007/BF01206656. JFM 23.0222.02

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.