A law of the iterated logarithm for the number of occupied boxes in the Bernoulli sieve

The Bernoulli sieve is an infinite occupancy scheme obtained by allocating the points of a uniform $[0,1]$ sample over an infinite collection of intervals made up by successive positions of a multiplicative random walk independent of the uniform sample. We prove a law of the iterated logarithm for the number of non-empty (occupied) intervals as the size of the uniform sample becomes large.