We develop, in the context of the boundary of a supercritical Galton–Watson tree, a uniform version of large deviation estimate on homogeneous trees to estimate almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot measure over a suitable set [Formula: see text]. As an application, we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of a fractal set related to covering number on the Galton–Watson tree.

中文翻译：

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Hausdorff 和 Mandelbrot 测量的包装尺寸

我们在超临界 Galton-Watson 树的边界的背景下，开发了对同质树的大偏差估计的统一版本，以几乎肯定地同时估计 Mandelbrot 测量的 Hausdorff 和包装维度在合适的集合上 [公式：见文本]。作为一个应用程序，我们几乎可以肯定地同时计算与高尔顿-沃森树上的覆盖数相关的分形集的 Hausdorff 和包装维数。