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

Abstract

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.