The rapid points of a complex oscillation

By considering a counting-type argument on Brownian sample paths, we prove a result similar to that of Orey and Taylor on the exact Hausdorff dimension of the rapid points of Brownian motion. Because of the nature of the proof we can then apply the concepts to so-called complex oscillations (or ‘algorithmically random Brownian motion’), showing that their rapid points have the same dimension.

💡 Research Summary

The paper revisits the classical problem of determining the Hausdorff dimension of the set of rapid points of a one‑dimensional Brownian motion, originally solved by Orey and Taylor (1974). A rapid point t for a parameter α∈(0,1) is defined by the limsup condition

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