Visible parts and slices of Ahlfors regular sets

Visible parts and slices of Ahlfors regular sets, Discrete Analysis 2024:17, 31 pp.

Let \(E\) be a set in \(\mathbb R^d\). Given a direction \(\theta\in S^{d-1}\), the visible part of \(S\) from that direction is the set of points that can be viewed from that direction unobstructed by other points of \(S\). We use the notation \(\hbox{Vis}_\theta(E)\) to denote that set.

We are interested in upper bounds on the Hausdorff dimension of \(\hbox{Vis}_\theta(E)\) relative to the Hausdorff dimension of \(E\). If \(E\) has dimension at most \(d-1\), a result of Mattila says that we have \(\dim_H(\hbox{Vis}_\theta(E))=\dim_H(E)\) for Lebesgue-almost all \(\theta\). A representative example is provided by a set \(E\) contained in a \((d-1)\)-dimensional hyperplane: for any direction \(\theta\) not parallel to the hyperplane, we have \(\hbox{Vis}_\theta(E)=E\).

On the other hand, when \(\dim_H(E)>d-1\), the set can no longer be laid out flat as in the last example, and one expects that most of \(E\) must be hidden from view from most directions. The visibility conjecture makes this intuition quantitative, asserting that we should in fact have \(\dim_H(\hbox{Vis}_\theta(E))\leq d -1\) in this case for almost all \(\theta\).

The first such result was proved by Orponen (J. Eur. Math. Soc. 2023): for any compact \(E\), we must always have
\[ \dim_H(\hbox{Vis}_\theta(E))\leq d-\frac{1}{50d} \]
for almost all \(\theta\).
The present paper improves Orponen’s bound to
\[ \dim_H(\hbox{Vis}_\theta(E))\leq d-\frac{1}{6} \]
for almost all \(\theta\), with the “gap” of size \(1/6\) independent of \(d\).

If \(d-1< \dim_H(E)\leq d-\frac{1}{6}\), the above bound is not meaningful, since we always have \(\hbox{Vis}_\theta(E)\subset E\) and therefore
\[\dim_H(\hbox{Vis}_\theta(E))\leq\dim_H(E).\]
The expectation based on the visibility conjecture is that the last inequality must in fact be strict, but, surprisingly, even this is still not known in general. However, if \(E\) is assumed to be Ahlfors-regular, then the author proves that
\[ \dim_H(\hbox{Vis}_\theta(E))\leq (1-2\alpha)s+\alpha(d-1) \]
for almost all \(\theta\), where \(\alpha=1-\frac{\sqrt{6}}{3}\). This provides strong evidence in support of the conjecture. The proofs are based on a Marstrand-like “slicing” theorem that is likely to be of independent interest to researchers in this area.