Dimensions of fractals related to languages defined by tagged strings in complete genomes

A representation of frequency of strings of length K in complete genomes of many organisms in a square has led to seemingly self-similar patterns when K increases. These patterns are caused by under-represented strings with a certain “tag”-string and they define some fractals in the K→∞ limit. The Box and Hausdorff dimensions of the limit set are discussed. Although the method proposed by Mauldin and Williams to calculate Box and Hausdorff dimension is valid in our case, a different and sampler method is proposed in this paper.