Hausdorff dimension and non-empty intersections with lines

Question

Let $A\subseteq [0,1]^d$, $d\geq 2$, a set with Hausdorff dimension $\operatorname{dim}_{\mathcal{H}}A=s$. What is the minimum $s$ (if any) which guarantee that $A$ has non-empty intersections with a positive fraction of lines passing through the origin? Thank you in advance for any suggestion.

Answer 1

This is true if the dimension of $A$ is strictly larger than $d-1$; on the other hand taking $A = \{ x^d = 0 \}$ shows that $s = d-1$ is not enough. To prove the first claim we use the co-area formula.

Remark. Generally, when working with Lipschitz functions for example, one has to be careful with the application of the co-area formula on irregular sets, which $A$ might well be. However, I believe that the formula as stated below is correct, because it 'conditions' on the values of a smooth function. I think the form below can be obtained from Fubini in spherical coordinates, integrating an indicator function.

A few preparations simplify the calculations.

First, pick $\delta > 0$ small enough that $s - \delta > d-1$. By definition of Hausdorff dimension, $\mathcal{H}^{s-\delta}(A) = \infty$, but we may consider a subset $B \subset A$ that has finite, but non-zero measure: $0 < \mathcal{H}^{s-\delta}(B) < \infty$. (If $B$ intersects a positive proportion of the lines through the origin, then $A$ does too.) To simplify the argument, we want to avoid the origin, so we may assume that $B \subset [0,1]^d \setminus \mathbf{B}^{d}(\rho)$ for some $\rho > 0$. Here $\mathbf{B}^d(\rho)$ is the closed ball of radius $\rho$ around the origin.

Let $f: x \in \mathbf{R}^d \setminus \{ 0 \} \mapsto \frac{x}{\lvert x \rvert}$. The level sets of $f$ are exactly the lines through the origin; we write $l_\omega \subset \mathbf{R}^d$ for the line directed by $\omega \in \mathbf{R}P^{d-1}$. The co-area formula essentially lets us 'condition' on these: \begin{equation} \int_{B} Jf \, \mathrm{d} \mathcal{H}^{s-\delta} = \int_{\mathbf{R}P^{d-1}} \mathcal{H}^{s-\delta-(d-1)}(B \cap l_\omega) \, \mathrm{d} \mathcal{H}^{d-1}, \end{equation} where $Jf = (\operatorname{det} Df \circ Df)^{1/2}$ is the Jacobian area change factor of $f$.

The function $f$ in question is simple enough to calculate this explicitly if you like, but for us it's enough to note that—because $f$ is $C^1$ away from the origin for example—there are constants $0 < c(d,\rho) < C(d,\rho)$ so that \begin{equation} c(d,\rho) < Jf < C(d,\rho) \quad \text{ on $[0,1]^d \setminus \mathbf{B}^d(\rho)$}. \end{equation}

We picked $B$ a subset of $[0,1] \setminus \mathbf{B}^d(\rho)$, so \begin{equation} c(d,\rho) \mathcal{H}^{s-\delta}(B) \leq \int_{\mathbf{R}P^{d-1}} \mathcal{H}^{s-\delta-(d-1)}(B \cap l_\omega) \, \mathrm{d} \mathcal{H}^{d-1} \leq C(d,\rho) \mathcal{H}^{s-\delta}(B). \end{equation} Therefore $\mathcal{H}^{s- \delta - (d-1)}(B \cap l_\omega)$ cannot vanish $\mathcal{H}^{d-1}$-a.e. everywhere on $\mathbf{R}P^{d-1}$, and in particular the set of $\omega \in \mathbf{R}P^{d-1}$ for which the intersection $B \cap l_\omega$ is non-empty has positive $\mathcal{H}^{d-1}$-measure.