The Fourier dimension of Brownian limsup fractals

TL;DR

This paper revises Kaufman's 1974 proof on the Fourier dimension of Brownian motion's rapid points, providing a corrected and extended approach to show these sets are Salem sets under certain conditions.

Contribution

It offers a corrected version of Kaufman's theorem and extends the method to demonstrate that certain rapid point sets are Salem sets for specific functions.

Findings

Corrected proof of Kaufman's theorem

Establishment of Fourier dimension equal to Hausdorff dimension

Extension to Salem sets for absolutely continuous functions

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. 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.