What time does it take for irrational rotations to hit an interval? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:23:34Z http://mathoverflow.net/feeds/question/110327 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110327/what-time-does-it-take-for-irrational-rotations-to-hit-an-interval What time does it take for irrational rotations to hit an interval? Antoine Levitt 2012-10-22T12:50:28Z 2012-10-29T03:04:33Z <p>Hi,</p> <p>Consider $\theta_n = (\theta_0 + n \theta) \mod 1$, $\theta$ being an irrational number, and $\theta_0$ an uniform random variable in $(0,1)$. Is there any estimates for the time it will take this process to hit $(0,\alpha)$ ? From the ergodic theorem I know that, if I denote $N(n)$ the number of times $\theta_n \in (0,\alpha)$, then $N(n)/n \to \alpha$. What I want to know is how much time it will take for this limit to be attained.</p> <p>Another way of framing this question is : is there any "central limit theorem" (or weakening thereof ; I'm mainly interested in guaranteed bounds for $P(N\geq 1)$) for ergodic processes? From what I've read, there is no general answer to this for a generic ergodic process and function f. There are some results that depend on $f$ being smooth, which it isn't here.</p> <p>The same question was asked on <a href="http://mathoverflow.net/questions/4411/quantitative-versions-of-ergodic-theorem" rel="nofollow">http://mathoverflow.net/questions/4411/quantitative-versions-of-ergodic-theorem</a>, but I haven't found anything there that relates to my question.</p> http://mathoverflow.net/questions/110327/what-time-does-it-take-for-irrational-rotations-to-hit-an-interval/110329#110329 Answer by Igor Rivin for What time does it take for irrational rotations to hit an interval? Igor Rivin 2012-10-22T13:46:06Z 2012-10-22T13:46:06Z <p>This is a very nice question! A lot of results (and references) are given in <a href="http://www-users.york.ac.uk/~zc3/download/circlimlaw.ps.gz" rel="nofollow">Zaq Coelho's "The loss of tightness of time distributions for homeomorphisms of the circle".</a></p> http://mathoverflow.net/questions/110327/what-time-does-it-take-for-irrational-rotations-to-hit-an-interval/110958#110958 Answer by LazyCat for What time does it take for irrational rotations to hit an interval? LazyCat 2012-10-29T03:04:33Z 2012-10-29T03:04:33Z <p>There is a theorem of Kesten, which roughly says, that if you take (\theta, \theta_0) random, and the number of times you hit (0, \alpha) in the first N iterations, subtract the expected N * \alpha, and normalize by \rho * ln(n), the result will converge to Cauchy distribution. This can be viewed as an analogue of CLT in this case. </p>