Some Probabilistic Remarks on the Boundary Version of Worpitzky’s Theorem

The old and celebrated theorem of Worpitzky states that all continued fractions (1) K n = 1 ∞ ( a n / 1 ) ,                   https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq3904.tif"/> where (2) | a n | ≤ 1 4 , n = 1 , 2 , 3 , ...                                           https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq3905.tif"/> converge [4]. Moreover, it can easily be proved that all approximants are in the closed disk (3) | w | ≤ 1 2 .                                     https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq3906.tif"/>