On the exact rate of convergence of frequencies of digits in self-similar sets

We study the exact rate of convergence of frequencies of digits of "normal" points of a self-similar set. Our results have applications to metric number theory. One particular application gives the following surprising result: there is an uncountable family of pairwise disjoint and exceptionally big subsets of R-d that do not obey the law of the iterated logarithm. More precisely, we prove that there is an uncountable family of pairwise disjoint and exceptionally big sets of points x in R-d-namely, sets with full Hausdorff dimension-for which the rate of convergence of frequencies of digits in the N-adic expansion of x is either significantly faster or significantly slower than the typical rate of convergence predicted by the law of the iterated logarithm.