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We examine a construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

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.

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In 1728, when the sixteen-year-old Hume, still apparently ‘at college’, was beginning, all unknown to his family, to turn his attention to philosophy, Edinburgh and Glasgow were swarming with earnest metaphysicians, many of them not much older than Hume himself. ‘It is well known’, the Ochtertyre papers relate, ‘that between the years 1723 and 1740 nothing was in more request with the Edinburgh literati, both laical and clerical, than metaphysical disquisitions’, and Locke, Clarke, Butler and Berkeley are mentioned as the chief subjects for debate. Moreover, it is clear enough from the records that this surge of intellectual interests was chiefly the work of a younger generation, wearied alike of Calvinist theology and of Jacobite politics. Indeed to begin with it was the students’ societies which took the lead, and a plain enough hint of their serious critical attack is given in one sour entry in the diary of the Calvinist minister Woodrow for 1726. ‘These student clubs are like to have a very ill influence; they declare against reading and cry up thinking.’

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