Let At denote the set of infinite sequences of effective dimension t. We
determine both how close and how far an infinite sequence of dimension s can be
from one of dimension t, measured using the Besicovitch pseudometric. We also
identify classes of sequences for which these infima and suprema are realized
as minima and maxima. When t < s, we find d(X,At) is minimized when X is a
Bernoulli p-random, where H(p)=s, and maximized when X belongs to a class of
infinite sequences that we call s-codewords. When s < t, the situation is
reversed.Comment: 28 pages, 3 figure