The generalized upper box dimension

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.