Some observations on the distribution of values of continued fractions

Abstract

Summary. There have been many studies of the values taken on by continued fractions K(a./1) when its elements are all in a prescribed set E. The set of all values taken on is the limit region V(E). It has been conjectured that the values in V(E) are taken on with varying probabilities even when the elements a. are uniformly distributed over E. In this article we present the first concrete evidence that this is indeed so. We consider two types of element regions: (A) E is an interval on the real axis. Our best results are for intervals [-p(1--p), p(1--p)], 0<p< 89 (B)E is a disk in the complex plane defined by E={z: lzl<p(1-p)},0<p< 89