Denote by x a random innnite path in the graph of Pascal's triangle (left and right turns are selected independently with xed probabilities) and by d n (x) the binomial coeecient at the n'th level along the path x. Then for a dense G set of in the unit interval, fd n (x)g is almost surely dense but not uniformly distributed modulo 1.

Normalized compression distance (NCD) is a parameter-free similarity measure based on compression. The NCD between pairs of objects is not sufficient for all applications. We propose an NCD of finite multisets (multiples) of objacts that is metric and is better for many applications. Previously, attempts to obtain such an NCD failed. We use the theoretical notion of Kolmogorov complexity that f...

Denote by x a random infinite path in the graph of Pascal’s triangle (left and right turns are selected independently with fixed probabilities) and by dn(x) the binomial coefficient at the n’th level along the path x. Then for a dense Gδ set of θ in the unit interval, {dn(x)θ} is almost surely dense but not uniformly distributed modulo 1.