On the Exact Dimension of Mandelbrot Measure

We develop, in the context of the boundary of a supercritical Galton-Watson tree, a uniform version of the argument used by Kahane (1987) on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set J. As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets E(α) of infinite branches of the boundary of the tree along which the averages of the branching random walk have a given limit point.

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Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).