Projection theorems for macroscopic intermediate dimensions

@misc{li2026projection,
  title = {{Projection theorems for macroscopic intermediate dimensions}},
  author = {Jinjun Li and Lifeng Xi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04395},
  eprint = {2610.04395},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04395},
}

TY  - UNPB
TI  - Projection theorems for macroscopic intermediate dimensions
AU  - Jinjun Li
AU  - Lifeng Xi
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04395
UR  - https://arxiv.org/abs/2610.04395
AB  - 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⊂R^d and m∈ {1,…,d−1}, we prove that the macroscopic intermediate dimensions of P_VE equal the corresponding intrinsic capacity profiles for almost every m-dimensional linear subspaces V⊂ R^d. One full-measure set works for both dimensions and every parameter θ∈[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.
ER  -