The generalized upper box dimension
Lipeng Wang, Wenxia Li
TL;DR
The paper extends fractal dimension theory to unbounded sets by introducing the generalized upper box dimension $\overline{\dim}_{GB}$, defined as $\limsup_{\theta\to 0} \dim_A^\theta F$ (equivalently $\lim_{\theta\to 0} \overline{\dim}_A^\theta F$). It proves that for bounded sets this generalization collapses to the classical upper box dimension, and that a modified variant equals the packing dimension, thereby connecting spectrum-based and classical dimensions. It further shows that $\overline{\dim}_{GB} F=0$ iff $\dim_{qA} F=0$, and that the upper spectrum being full (or zero) is equivalent to corresponding properties of the Assouad spectrum, linking these spectral notions. Overall, the work builds a coherent framework tying generalized box dimensions to spectrum-based dimensions, enabling analysis of unbounded and irregular sets within fractal geometry.
Abstract
We introduce the generalized upper box dimension which is defined for any set, whether the set is bounded or unbounded. We study basic properties of the generalized upper box dimension. We prove that the generalized upper box and upper box dimensions coincide for bounded sets. Furthermore, we also show that the modified generalized upper box dimension equals the packing dimension. So the generalized upper box dimension can be seen as a reasonable generalization of the upper box dimension. As an application, we prove the generalized upper box dimension is zero if and only if the quasi-Assouad dimension is zero. We also show that the upper spectrum is of full dimension is equivalent to the Assouad spectrum is of full dimension and the upper spectrum is zero is equivalent to the Assouad spectrum is zero.