Visibility of unrectifiable planar sets

Matt Bond, Josh Zahl and I have just completed a new paper “Quantitative visibility estimates for unrectifiable sets in the plane,” now available on the arXiv. This post is an informal introduction to the paper; for more details, you will need to download the actual article.

There are several questions known as “visibility problems”, and the one we address is the following. We are given a compact set E in the plane, and a point a not in E. Define P_a to be the radial projection from a:

P_a(x) = \frac{x-a}{|x-a|}

Then P_a(E) is the set of angles at which E is visible from a. Our “visibility problem” is then to estimate the size of |P_a(E)|, or equivalently, the proportion of the part of the field of vision that E takes up for an observer situated at a.

One class of sets that we will study is 1-dimensional unrectifiable self-similar sets. A good example to keep in mind is the “4-corner set,” constructed via a Cantor iteration as follows. Start with a square, divide in into 16 congruent squares, and keep the 4 small squares at the corners, discarding the rest. Repeat the same procedure for each of the 4 small surviving squares, then iterate the construction. The first and second stage of the iteration are shown below.

4corners1

4corners2

We will use K_n for the n-th iteration of this set, and K for the Cantor set K = \bigcap_{n=1}^\infty K_n.

What can we say about the visibility of K from points a in the plane? We will assume that a \notin K, so as to avoid trivial debates over whether a point is visible from itself. We will be asking this question in terms of the size of P_a(K), as expressed in terms of its Lebesgue measure and/or Hausdorff dimension.

It’s easy to see that for every a, the set P_a(K) has dimension at least 1/2. This is because at least one of the sides of the “outer square” is visible from a at a non-zero angle, and K intersects each of these sides in a set of dimension 1/2. Can we say more than that? What about the upper bounds? There are directions (e.g. slope 1/2) where the linear projection of K has positive Lebesgue measure; can that happen for radial projections?

Upper bounds. Using projective transformations (to convert linear projections to radial projections), it is easy to deduce from Marstrand’s projection theorem that a purely unrectifiable 1-dimensional set E is invisible from almost every point: |P_a(E)|=0 for Lebesgue-a.e. a\in {\bf R}^2. Marstrand (1954) proved the stronger result that the set of exceptional points from which E is visible can have Hausdorff dimension at most one, and gave an example showing that a 1-dimensional set of exceptional points is indeed possible.

That does not, however, happen for self-similar sets such as K. Simon and Solomyak (2006) proved that 1-dimensional unrectifiable self-similar sets are in fact invisible from every point: we have |P_a(K)|=0 for all a\in {\bf R}^2. The underlying principle is that, for self-similar sets, radial projections are better behaved than e.g. linear projections because they have averaging over angles built into them already. This will be a recurring theme in this work.

Lower bounds. Using Marstrand’s theorem and projective transformations again, it’s easy to see that P_a(E) has dimension 1 for Lebesgue-a.e. a in the plane. There’s no lower bound on how small P_a(E) can be for the exceptional points. For example, we could take E to be a product of two Cantor sets in polar coordinates r and \theta, of dimension 1-\alpha and \alpha respectively; then P_0(E) has dimension \alpha, which could be any number between 0 and 1, endpoints included. (In an extreme case, E could lie on a line, and then P_a(E) would consist of a single point for every a on the same line. But then E would not be unrectifiable.)

For self-similar sets, however, the stronger statement is true that P_a(E) has Hausdorff dimension 1 for every a. This is a consequence of recent deep results of Hochman and Hochman-Shmerkin. The argument was pointed out to us by Mike Hochman, and we’re grateful to him for allowing us to include it in our paper.

Quantitative upper bounds. This brings us to the main subject of our paper, namely quantitative visibility bounds. We start with upper bounds for self-similar sets such as K. Recall that K_n is the n-th Cantor iteration of K. The result of Simon and Solomyak implies that

|P_a(K_n)| \to 0 as n \to \infty.

What about the rate of decay? It turns out that this can indeed be quantified. In the case of K_n, we prove that

(1) \ \ |P_a(K_n)| \leq C_p ( \log n)^{-p},

for any p < 1/12. This is a consequence of the following result. Let Fav (E) be the Favard length of E, that is, the average (with respect to angle) length of its linear projections. Then if E is a self-similar set defined by homotheties with equal contraction ratios, and E_n is its n-th iteration, we have

(2) \ \  |P_a(E_n)| \leq C_1 \sqrt{ Fav(E_{C_2\log n})}

with the constants uniform for a in a fixed compact set disjoint from E. Combining this with the known Favard length estimate for K_n (due to Nazarov, Peres and Volberg), we get (1). For general self-similar sets defined by homotheties with equal contraction ratios, we can instead use a result of Bond and Volberg to estimate the right side of (2), and get that

(3) \ \ |P_a(E_n)| \leq C_1 \exp (- C_2 \sqrt{ \log \log n} )

The constants in (1)-(3) can be taken to be independent of a if a is restricted to a compact set disjoint from E. If the self-similar set is defined by more general similitudes, with rotations and/or different contraction ratios, we can still get a weaker analogue of (2), but unfortunately no effective Favard length estimates are available in this case.

Quantitative lower bounds. Here, we consider a wider class of “discrete unrectifiable 1-sets” defined in the paper. This includes \delta-neighbourhoods of self-similar sets as above, but also other more general examples such as diffeomorphic images of self-similar sets. For example, \delta-neighbourhoods of 1-dimensional unrectifiable product Cantor sets in polar coordinates fall in this category.

If E_\delta is discretized on scale \delta (essentially, a union of \delta-balls), we clearly have |P_a(E_\delta )|\geq \delta for all points a. In general, we cannot say much more about visibility from any individual point: it could be as small as \delta^{1-\epsilon} for any \epsilon>0. (This can be seen by considering the above example of product sets in polar coordinates, where the “angular” dimension is \epsilon.) However, it is reasonable to expect that the set of points of low visibility should be small.

In that regard, we have the following result. Let \lambda \in (0,1), and let F\subset \mathbb{R}^2 be a compact set. Then

(4) \ \ |\{a\in F \colon |P_a(E_\delta)| < \lambda \}|  \leq C_1| \log \delta|^{C_2} \lambda^{2}.

If moreover \lambda < \delta^{1/2-\epsilon_0} for some \epsilon_0 small enough, then there is an \epsilon_1>0 such that

(5) \ \ |\{a\in F \colon |P_a(E_\delta)| < \lambda \}|  \leq C_1| \log \delta|^{C_2} \lambda^{2+\epsilon_1}.

We also have analogues of (4) and (5) for sets of dimension \alpha \in (0,1); see the article for the definitions and numerology. The first estimate (4) is proved using L^2 incidence combinatorics; (5) starts similarly, but the additional key ingredient is Bourgain’s “discretized Marstrand projection estimate”.

In the case of self-similar sets such as K, we identify K_n with a \delta-neighbourhood of K (in this case, \delta=4^{-n}). The dimensionality result mentioned above implies that for every a \notin K, and for every \epsilon>0, we have

(6) \ \ |P_a(K_n)| \geq C(a,\epsilon) 4^{-n\epsilon}.

Pointwise, (6) is stronger than (4) or (5). On the other hand, (6) is not uniform in a, therefore does not provide any estimates on the size of sets in (4), (5).

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