Box dimension prints

RESUMO

We study lower and upper box dimension prints for bounded subsets of \(\mathbb R^n\), defined by weighted covering numbers for independently oriented rectangular boxes with prescribed ordered side-length bounds. The limits range over all eccentricities, including unbounded aspect ratios. For every non-empty bounded set, we identify the closure of the lower print with the intersection of the half-spaces determined by its lower eccentricity profile, without any uniformity assumption. The profile equals the support function of this closure if and only if it is subadditive. A planar product example has distinct lower and upper profiles on every ray and an explicitly computable lower-print closure. We also prove that both prints are invariant under nonsingular projective transformations on compact sets avoiding the pole hyperplane. Uniform anisotropic covering estimates determine both prints, including their boundary points, for non-degenerate curves of type \((1,\ldots,n)\), their Ahlfors regular parameter subsets, and higher-dimensional spheres. Finally, local covering-count and product-measure criteria identify the lower box print with the Hausdorff dimension print, while a reciprocal-sequence example shows that this inclusion can be strict.