Diophantine properties of Brownian motion: recursive aspects

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.