On the approximation by Lüroth Series

Abstract

Let x 2 (0; 1] and pn=qn; n 1 be its sequence of Luroth Series convergents. Dene the approximation coecients n = n(x) by n = qnx. In [BBDK] the limiting distribution of the sequence (n)n1 was obtained for a.e. x using the natural extension of the ergodic system underlying the L? uroth Series expansion. Here we show that this can be done without the natural extension. We also will get a bound on the speed of convergence. Using the natural extension we will study the distribution for a.e. x of the sequence (n; n+1)n1 and related sequences like (n + n+1)n1. It turns out that for a.e. x the sequence (n; n+1)n1 is distributed according to a continuous singular distribution function G. Furthermore we will see that two consecutive 's are positively correlated