A Polynomial Time Algorithm for the Hausdorff Dimension of Continued Fraction Cantor Sets Academic Article uri icon

abstract

  • For any finite set A of positive integers, let EA := {α∈ (0, 1): α is irrational, and every partial quotient in the (infinite) simple continued fraction expansion of α is an element of A}. For sets A with fewer than two elements, EA is uninteresting. For A≥2, EA is a kind of Cantor fractal dust, with a Hausdorff dimension (dim EA) between 0 and 1. This work presents an algorithm which, given a finite set A of between 2 and N positive integers 2N, determines dim EA to within ±2-N using O(N7) elementary bit operations. There is also a convenient implementation of the algorithm in Mathematica® code, together with a small table and some conjectures. © 1996 Academic Press, Inc.

author list (cited authors)

  • Hensley, D.

citation count

  • 36

publication date

  • May 1996