A Polynomial Time Algorithm for the Hausdorff Dimension of Continued Fraction Cantor Sets Original Research Article

For any finite setAof positive integers, letEA :={αset membership, variant(0, 1):αis irrational, and every partial quotient in the (infinite) simple continued fraction expansion ofαis an element ofA}. For setsAwith fewer than two elements,EAis uninteresting. For Agreater-or-equal, slanted2,EAis 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 setAof between 2 andNpositive integers 2N, determines dim EAto within ±2−NusingO(N7) elementary bit operations. There is also a convenient implementation of the algorithm in Mathematica® code, together with a small table and some conjectures.