Orthogonal Projections and Assouad Dimension: Resolving Fraser's Questions
Abstract
We examine two questions on exceptional orthogonal projections for Assouad dimension. Let 1≤ k<d and 1≤ q<d. For a set F contained in a q-dimensional subspace W⊂Rd and containing a nonempty relatively open subset of W, we compute both the Hausdorff and packing dimensions of the exceptional k-planes as $k(d-k)-|q-k|-1$. In particular, a square in R3 has a one-dimensional set of exceptional plane projections. This disproves the bound k(d−1−k) in Question 17.8.1 of Fraser's book as stated, for every 2≤ k<d. The planar packing question, Question 17.8.2, is already answered affirmatively by Wu's countability theorem. Using that theorem, we also describe the distinction between planar sets of Assouad dimension at most $1$ and those of dimension greater than $1$ when they are embedded in higher-dimensional spaces. The computations use elementary Grassmannian geometry; we make no claim of a new planar projection theorem or of an optimal bound for arbitrary higher-dimensional sets.
Related
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BibTeX
@misc{selmi2026orthogonal,
title = {{Orthogonal Projections and Assouad Dimension: Resolving Fraser's Questions}},
author = {Bilel Selmi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23295},
eprint = {2610.23295},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23295},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Orthogonal Projections and Assouad Dimension: Resolving Fraser's Questions
AU - Bilel Selmi
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23295
UR - https://arxiv.org/abs/2610.23295
AB - We examine two questions on exceptional orthogonal projections for Assouad dimension. Let 1≤ k<d and 1≤ q<d. For a set F contained in a q-dimensional subspace W⊂R^d and containing a nonempty relatively open subset of W, we compute both the Hausdorff and packing dimensions of the exceptional k-planes as $k(d-k)-|q-k|-1$. In particular, a square in R^3 has a one-dimensional set of exceptional plane projections. This disproves the bound k(d−1−k) in Question 17.8.1 of Fraser's book as stated, for every 2≤ k<d. The planar packing question, Question 17.8.2, is already answered affirmatively by Wu's countability theorem. Using that theorem, we also describe the distinction between planar sets of Assouad dimension at most $1$ and those of dimension greater than $1$ when they are embedded in higher-dimensional spaces. The computations use elementary Grassmannian geometry; we make no claim of a new planar projection theorem or of an optimal bound for arbitrary higher-dimensional sets.
ER -