  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
$\alpha$-Cantor set, $\alpha \in (0,1)$, with positive probability if and only
if $\alpha \neq 1/2$. 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.
