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Estimates of Kolmogorov complexity in approximating cantor sets

Our aim is to quantify how complex a Cantor set is as we approximate it better and better. We formalize this by asking what is the shortest program, running on a universal Turing machine, which produces this set at the precision ε in the sense of Hausdorff distance. This is the Kolmogorov complexity of the approximated Cantor set, which we call the ‘ε-distortion complexity’. How does this quantity behave as ε tends to 0? And, moreover, how this behaviour relates to other characteristics of the Cantor set? This is the subject of this work: we estimate this quantity for several types of Cantor sets on the line generated by iterated function systems and exhibit very different behaviours. For instance, the ε-distortion complexity of most Ck Cantor sets is proven to behave as ε−D/k, where D is its box counting dimension.