Abstract

It is well-known that the notion of randomness, suitably refined, goes a long way in dealing with the tension between the ``incompatability of shortest descriptions and of effecting the most-economical algorithmical processing” Manin(2006). In this work, we continue to explore this interplay between short descriptions and randomness in the context of Brownian motion and its associated geometry. In this way one sees how random phenomena associated with the geometry of Brownian motion, are implicitly enfolded in each real number which is complex in the sense of Kolmogorov. These random phenomena range from fractal geometry, Fourier analysis and non-classical noises in quantum physics. In this talk we shall discuss countable dense random sets as the appear in the theory of Brownian motion in the context of algorithmic randomness. We shall also discuss applications to Fourier analysis. In particular, we also discuss the images of certain $Pi_2^0$ perfect sets of Hausdorff dimension zero under a complex oscillation (which is also known as an algorithmically random Brownian motion). This opens the way to relate certain non-classical noises to Kolmogorov complexity. For example, the work of the present work enables one to represent Warren’s splitting noise directly in terms of infinite binary strings which are Kolmogorov-Chaitin-Martin-Löf random.