[Paper Review] Orthogonal projections of discretized sets

Weikun He|arXiv (Cornell University)|Oct 2, 2017

Advanced Topology and Set Theory12 references5 citations

TL;DR

This paper generalizes Bourgain's discretized projection theorem to higher-rank orthogonal projections in $$\mathbb{R}^n$$, establishing that for a discretized set $A$ of dimension $\alpha$, its projection onto most $m$-dimensional subspaces has dimension at least $\frac{m}{n}\alpha - \epsilon$, under non-concentration assumptions on $A$ and the distribution of subspaces. The result extends fractal projection theorems to higher codimensions using covering numbers and geometric measure theory.

ABSTRACT

We generalize Bourgain's discretized projection theorem to higher rank situations. Like Bourgain's theorem, our result yields an estimate for the Hausdorff dimension of the exceptional sets in projection theorems formulated in terms of Hausdorff dimensions. This estimate complements earlier results of Mattila and Falconer.

Motivation & Objective

• To extend Bourgain's discretized projection theorem from rank-one to higher-rank orthogonal projections in $\mathbb{R}^n$.
• To provide quantitative estimates on the Hausdorff dimension of exceptional projection directions in terms of covering numbers at scale $\delta$.
• To establish a higher-rank analog of the discretized projection theorem that controls the size of projections under non-concentration assumptions on both the set $A$ and the measure $\mu$ on the Grassmannian.
• To derive fractal geometric consequences, including a projection theorem in terms of Hausdorff dimension for analytic sets of dimension $\alpha$.

Proposed method

• Uses covering numbers $\mathcal{N}_\delta(A)$ to measure the size of discretized sets at scale $\delta$, replacing Lebesgue or Hausdorff measures.
• Imposes a Frostmann-type non-concentration condition (2) on $A$, ensuring $A$ does not concentrate at small scales.
• Imposes a non-concentration condition (3) on the measure $\mu$ on $\operatorname{Gr}(\mathbb{R}^n, m)$, requiring $\mu(\mathcal{V}_{\measuredangle}(W,\rho)) \leq \delta^{-\epsilon}\rho^\kappa$ for all $W$ and $\rho \geq \delta$.
• Introduces the angular distance $\operatorname{d_{\measuredangle}}(V,W)$ to define neighborhoods $\mathcal{V}_{\measuredangle}(W,\rho)$ of Schubert cycles in the Grassmannian.
• Applies the Łojasiewicz inequality to control the covering number of $\mathcal{V}_{\measuredangle}(W,\rho)$ in terms of $\rho^{1/C}$, enabling dimension estimates.
• Uses Brudnyi's Remez-type inequality for real analytic functions on manifolds to control the size of exceptional parameter sets in restricted families of projections.

Experimental results

Research questions

• RQ1 What is the minimal dimension of projections of a discretized set $A \subset \mathbb{R}^n$ onto $m$-dimensional subspaces, under non-concentration assumptions?
• RQ2 How can Bourgain's discretized projection theorem be extended from rank-one to higher-rank projections in $\mathbb{R}^n$?
• RQ3 What is the maximal Hausdorff dimension of the set of exceptional directions $V$ for which $\dim_H(\pi_V(A)) \leq \frac{m}{n}\alpha + \epsilon$?
• RQ4 How does the non-concentration of the measure $\mu$ on the Grassmannian affect the size of projections of $A$?
• RQ5 What is the size of the exceptional parameter set in a family of projections $V(t)$, when the map $t \mapsto V(t)$ is real analytic?

Key findings

• For any $\epsilon > 0$, there exists $\delta_0 > 0$ such that for all $\delta < \delta_0$, the projection $\pi_V(A')$ of any subset $A' \subset A$ with $\mathcal{N}_\delta(A') \geq \delta^\epsilon \mathcal{N}_\delta(A)$ satisfies $\mathcal{N}_\delta(\pi_V(A')) \geq \delta^{-\frac{m}{n}\alpha - \epsilon}$ for a $\mu$-large set $\mathcal{D} \subset \operatorname{Gr}(\mathbb{R}^n, m)$ with $\mu(\mathcal{D}) \geq 1 - \delta^\epsilon$.
• The exceptional set of directions $V$ for which $\dim_H(\pi_V(A)) \leq \frac{m}{n}\alpha + \epsilon$ cannot support any nonzero measure $\mu$ satisfying $\mu(\mathcal{V}_{\measuredangle}(W,\rho)) \leq \rho^\kappa$ for all $\rho > 0$, thus controlling the size of exceptional sets in terms of Hausdorff dimension.
• The dimension of the exceptional parameter set in a real analytic family $V(t)$ of $m$-planes is at most $p - 1 + d\kappa$, where $d$ is the degree of the analytic function $f(W,t) = \operatorname{d_{\measuredangle}}(V(t),W)^2$, under transversality assumptions.
• The non-concentration condition on $\mu$ is quantified via $\delta^{-\epsilon}\rho^\kappa$, allowing the theorem to hold even when $\mu$ is slightly concentrated, as long as the concentration is controlled up to scale $\delta^\epsilon$.
• The result is sharp in the sense that the exponent $\frac{m}{n}\alpha$ is the expected dimension for projections of a set of dimension $\alpha$, and the $\epsilon$-loss is unavoidable under the given assumptions.
• The proof relies on the Łojasiewicz inequality to relate the covering number of $\mathcal{V}_{\measuredangle}(W,\rho)$ to $\rho^{1/C}$, and on Brudnyi's Remez-type inequality to control the size of exceptional parameter sets in analytic families.