Abstract
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. Some of the original arguments presented by Kaufman are not clear, hence this paper presents a new version of the construction and incorporates some recent results in order to establish a slightly weaker version of Kaufman’s theorem. This weaker version still implies that the Fourier and Hausdorff dimensions of Brownian rapid points are equal. 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.
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