Diophantine properties of Brownian motion: recursive aspects
Willem L. Fouche
TL;DR
This paper investigates the diophantine properties of algorithmically random Brownian motion using Fourier analysis, focusing on the construction of special sets and the properties of local minimizers.
Contribution
It introduces new methods to analyze the diophantine nature of Brownian motion and constructs definable perfect sets from Martin-Löf random reals.
Findings
Constructed linearly independent perfect sets from random reals.
Analyzed the diophantine properties of local minimizers of Brownian motion.
Connected Fourier analysis with diophantine properties of stochastic processes.
Abstract
We use recent results on the Fourier analysis of the zero sets of Brownian motion to explore the diophantine properties of an algorithmically random Brownian motion (also known as a complex oscillation). We discuss the construction and definability of perfect sets which are linearly independent over the rationals directly from Martin-L\"of random reals. Finally we explore the recent work of Tsirelson on countable dense sets to study the diophantine properties of local minimisers of Brownian motion.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Advanced Topology and Set Theory · Mathematical Dynamics and Fractals