Discrete intermediate dimensions: endpoints, spectra and quasi-isometric invariance

Abstract

We introduce upper and lower discrete intermediate dimensions for unbounded subsets of Euclidean space and extend them to metric spaces. The definitions restrict the covering radii within a scale-dependent range of dyadic shells and take a supremum over an auxiliary parameter. The resulting dimensions are monotone in the parameter and recover the Barlow--Taylor discrete Hausdorff dimension at zero. We prove continuity at every positive parameter, with a sharp quantitative bound, and give an example with a jump at zero. At the upper endpoint, the upper dimension equals upper mass dimension for uniformly discrete Euclidean sets; the analogous lower mass identity fails. We establish a mass distribution principle and prove quasi-isometric invariance under a uniform coarse packing condition. For spaces with uniform polynomial volume growth, we construct clustered subsets attaining the continuity bound. Independent thinning truncates these spectra at the resulting mass dimension, almost surely for all parameters simultaneously. Applications include the discrete Heisenberg group with its word metric and the ranges of simple symmetric random walks.