a1 University of Illinois, Urbana, Illinois, U.S.A.
In this paper X(t) denotes Brownian motion on the line 0 ≤ t < ∞, E is a compact subset of (0, ∞) and F a compact subset of ( -∞, ∞). Then δ(E, F) is the supremum of the numbers c such that
In [4, 5] some lower and upper bounds for δ were found in terms of dim E and dim F, and it seemed possible that δ(E, F) could be determined entirely by these constants; this much is false, as the examples in our last paragraph will demonstrate. Here we shall show that δ(E, F) depends on a certain metric character η(E × F). However, η is not calculated relative to the Euclidean metric: the set F must be compressed to compensate for the oscillations of most paths X(t). Fortunately, η(E × F) can be calculated for a large enough class of sets E and F, by means of sequences of integers, to test any conjectures (and disprove most of them). In passing from η to δ we present a slight variation of Frostman's theory [3II].
(Received June 21 1972)