Fourier spectra of measures associated with algorithmically random Brownian motion

TL;DR

This paper investigates the asymptotic decay of Fourier transforms of measures supported on fractal sets transformed by algorithmically random Brownian motion, linking computability, fractal geometry, and harmonic analysis.

Contribution

It establishes optimal decay rates for Fourier transforms of measures on fractals under algorithmically random Brownian motion, using computability and potential theory.

Findings

Fourier transforms exhibit optimal decay relative to Hausdorff dimension.

Computability conditions on fractals influence Fourier decay rates.

Characterization of algorithmically random Brownian motion via Kolmogorov complexity.

Abstract

In this paper we study the behaviour at infinity of the Fourier transform of Radon measures supported by the images of fractal sets under an algorithmically random Brownian motion. We show that, under some computability conditions on these sets, the Fourier transform of the associated measures have, relative to the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity. The argument relies heavily on a direct characterisation, due to Asarin and Pokrovskii, of algorithmically random Brownian motion in terms of the prefix free Kolmogorov complexity of finite binary sequences. The study also necessitates a closer look at the potential theory over fractals from a computable point of view.