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A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets. (English) Zbl 0858.11039
For each $A\subset \bbfN$, the set $E_A \subset (0,1)$ is defined to be those irrationals $\alpha$ with continued fraction expansion given by $\alpha = [0;a_1,a_2, \dots]$ with $a_i\in A$. When the cardinality of $A$ is finite, such $\alpha$ are badly approximable and $E_A$ is of measure 0. When $A$ has just one element, $E_A$ consists of a single quadratic irrational. When $2\le \text{Card} A \le N$, $E_A$ is a Cantor-type set with a `fractal dust’ structure. The author presents a polynomial time algorithm for determining the Hausdorff dimension of $E_A$ to within $\pm 2^{-N}$ using $O(N^7)$ operations. An implementation of the code in Mathematica is included. The Hausdorff dimension is 1/2 the unique value for which a function $\lambda$ related to a general zeta function is unity. In an interesting application of operator theory, the method uses results concerning the spectrum of an operator associated with the invariant measure for the continued fraction process. A number of conjectures suggested by calculations are included.

11K50Metric theory of continued fractions
11Y99Computational number theory
47A25Spectral sets
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