  We establish a relationship between the online mistake-bound model of
learning and resource-bounded dimension. This connection is combined with the
Winnow algorithm to obtain new results about the density of hard sets under
adaptive reductions. This improves previous work of Fu (1995) and Lutz and Zhao
(2000), and solves one of Lutz and Mayordomo's "Twelve Problems in
Resource-Bounded Measure" (1999).
