It was shown by Antunovi\'{c}, Burdzy, Peres, and Ruscher that a Cantor
function added to one-dimensional Brownian motion has zeros in the middle
α-Cantor set, α∈(0,1), with positive probability if and only
if α=1/2. We give a refined picture by considering a generalized
version of middle 1/2-Cantor sets. By allowing the middle 1/2 intervals to vary
in size around the value 1/2 at each iteration step we will see that there is a
big class of generalized Cantor functions such that if these are added to
one-dimensional Brownian motion, there are no zeros lying in the corresponding
Cantor set almost surely.Comment: 19 pages, improved Theorem 3.