Projection theorems for macroscopic intermediate dimensions

Abstract

We introduce lower and upper macroscopic intermediate dimensions of unbounded subsets of Euclidean space by combining covering contributions from all dyadic annuli beyond a given scale. For every unbounded Borel set $E\subset {\mathbb{R}}^d$ and $m\in \{1,\dots ,d-1\}$, we prove that the macroscopic intermediate dimensions of $P_{V}E$ equal the corresponding intrinsic capacity profiles for almost every $m$-dimensional linear subspaces $V\subset {\mathbb{R}}^d$. One full-measure set works for both dimensions and every parameter $\theta \in [0,1]$. The proof combines a covering--capacity comparison with an exponent gap, fractional-moment estimates for projected coverings, and capacity-weighted retention of source shells. The dimensions recover macroscopic Hausdorff dimension at parameter zero, are invariant under quasi-isometries, and are continuous at positive parameters. The upper macroscopic intermediate dimension agrees with the upper discrete intermediate dimension introduced in \cite{LXZ}, whereas the lower one can differ. We compute the projected dimensions of bounded-base discrete digit Moran sets, including the self-similar case obtained when the bases and digit sets are constant, and give applications to Lipschitz graphs and arithmetic sumsets.