  Robert Kaufman's proof that the set of rapid points of Brownian motion has a
Fourier dimension equal to its Hausdorff dimension was first published in 1974.
A study of the proof of the original paper revealed several gaps in the
arguments and a slight inaccuracy in the main theorem. This paper presents a
new version of the construction and incorporates some recent results in order
to establish a corrected version of Kaufman's theorem. The method of proof can
then be extended to show that functionally determined rapid points of Brownian
motion also form Salem sets for absolutely continuous functions of finite
energy.
