Orthogonal Projections and Assouad Dimension: Resolving Fraser's Questions

Abstract

We examine two questions on exceptional orthogonal projections for Assouad dimension. Let $1\le k<d$ and $1\le q<d$. For a set $F$ contained in a $q$-dimensional subspace $W\subset\R^d$ 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 $2\le k<d$. 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.