In this paper, we investigate how many real numbers can be well approximated by their convergents in the Engel expansions. Furthermore, the relative growth rate of convergence speed of convergents in the Engel expansion of an irrational number is studied to the rate of growth of its digits. The Hausdorff dimension of exceptional sets of points with a given relative growth rate is established.

中文翻译：

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ENGEL 扩展和 HAUSDORFF 维数的相对收敛速度

在本文中，我们研究了有多少实数可以很好地近似于它们在恩格尔展开式中的收敛性。此外，研究了无理数的恩格尔展开式中收敛子收敛速度的相对增长率与其位数的增长率。建立了具有给定相对增长率的异常点集的 Hausdorff 维数。