标题： 广义分形巢的盒计数维数 显示英文标题

标题： Box counting dimensions of generalised fractal nests

摘要： 分形巢是定义为单位$n$-球的并集，这些球被一个序列的$k^{-\alpha}$缩放，对于某些$\alpha>0$。 在本文中，我们将这一概念推广到这些球的子集，并找到它们的盒计数维数的公式。 我们引入了一些参数化的新型分形巢类，并将这些结果应用于根据这些参数计算维数。 我们还表明这些维数可以从数值上观察到。 这些结果激发了进一步的研究，这可能解释盒计数维数在巢型分形中的不可直观行为，以及更一般地，盒计数维数与豪斯多夫维数不同的集合类。 显示英文摘要

摘要： Fractal nests are sets defined as unions of unit $n$-spheres scaled by a sequence of $k^{-\alpha}$ for some $\alpha>0$. In this article we generalise the concept to subsets of such spheres and find the formulas for their box counting dimensions. We introduce some novel classes of parameterised fractal nests and apply these results to compute the dimensions with respect to these parameters. We also show that these dimensions can be seen numerically. These results motivate further research that may explain the unintuitive behaviour of box counting dimensions for nest-type fractals, and in general the class of sets where the box-counting dimension differs from the Hausdorff dimension.