Intersection of Cantor Sets

I will write about an interesting talk given by Pablo Shmerkin.

Main goal: computing dim(A\cap B) for A,B\subset \mathbb{R} with dynamics properties.

Theorem [Marstrand, 1954] E\subset\mathbb{R}^2 be a Borelian set. Given \theta\in (0,\pi), we define L_{\theta} to be the family of all the lines on the plane with slope \theta. Then for almost all l\in L_{\theta} we have that dim(E\cap l)\leq \max (dim(E)-1,0).

The graph G of a Brownian motion on the plane has dim(G)=\frac{3}{2}, however, vertical lines have intersection of HDIM equal to zero.

Theorem [Marstrand, 1954] E\subset\mathbb{R}^2 Borelian set. ess \mbox{ } \sup dim(E\cap l)=\max (dim(E)-1,0).

We can obtain the following corollaries for the HDIM of the intersection of subsets A,B\subset \mathbb{R}.

Corollary Let A,B\subset\mathbb{R} be Borelian set. Then for Leb almost all t we have that dim(A\cap (B+t))\leq \max (dim(A\times B)-1,0).

Remember that dim(A\times B)\geq dim(A)+dim(B), and that there is equality if dim(A)=\overline{dim_{BOX}(A)}. Therefore, we have the following corollaries.

Corollary If A\subset\mathbb{R} is a Borelian set such that dim(A)=\overline{dim_{BOX}(A)}, then for Leb almost all t we have that dim(A\cap (B+t))\leq \max (dim(A)+dim(B)-1,0).

And from the second theorem of Marstrand, we have the following corollary.

Corollary ess \mbox{ } \sup dim(A\cap g(B))=\max (dim(A\times B)-1,0), where g:\mathbb{R}\to\mathbb{R} is lineal.

Some examples.

There exists compact A\subset \mathbb{R}, dim(A)=1 such that the cardinality of A\cap(A+t)\leq 1 such that dim(A\cap(A+t))=0 and \max (dim(A)+dim(B)-1,0)=1.

For all s\in[0,1], \exists A\subset \mathbb{R}, dim(A)=s and dim(A\cap (A+t))=s for all t\in\mathbb{R}. [Falconer example]

Furstenberg asked what happened if A,B were invariant for the map T_p(x)=px \mod1. Motivated by this questions, let us consider first the classic Cantor set C\subset \mathbb{R}, whose dim(C)=\overline{dim_{BOX}(C)}=\frac{\log 2}{\log 3}. Applying directly Marstrand’s theorem we obtain the bound dim(C\cap (C+t))\leq 2\frac{\log 2}{\log 3}-1\approx 0.262 for almost all t. There is however a better bound by Hawkes.

Theorem [Hawkes, 1975] dim(C\cap (C+t))= \frac{1}{3}\frac{\log 2}{\log 3}\approx 0.21 for almost all t.

Kenyon and Peres generalized further in 1991 the result to the case of p\geq 2, D_1,D_2\subset \{0,1,\ldots,p-1\}, A_i=\{\sum x_j p^{-j}: x_j\in D_i\}. Notice that C=\{\sum x_j 3^{-j}: x_j\in \{0,2\}\}.

Theorem [R. Kenyon and Y. Peres, 1991] For almost all t\in [0,1] dim(A_1\cap (A_2+t))= \frac{\lambda}{\log p}. Where \lambda= greater Lyapunov exponent for the product of iid matrices M_r, where M_r(i,j)=\# (D_1+i+r)\cap (D_2+j+p), r=0,\ldots, p-1, 0\leq i,j\leq 1.

Barany and Rams in 2014 considered the case of self-similar sets E\subset [0,1]^2 that are obtained by a generalization of the set C\times C (ex. consider the unit square [0,1]?2, divide it in 9 squares of side 1/3, numerate the squares by \{1,2,\ldots,9\}=\{0,1,3-1\}^2. Choose 4 squares, say \Omega=\{1,2,5,9\}, delete the rest of the squares and repeat the procedure on each of the 4 squares left, etc.).

Theorem [B. Barany and M. Rams, 2014] Let E\subset \mathbb{R}^2 a set constructed doing the described process where \Omega\subset \{0,1,\ldots,p-1\}^2, p\nmid \#\Omega and \#\Omega>p. If \tan \theta \in \mathbb{Q}, then \exists \alpha(\theta),\beta(\theta) with \alpha(\theta)>dim(E)-1>\beta(\theta), dim(E\cap l)=\alpha(\theta) for \nu-almost all l\in L_{\theta}, where \nu is a natural measure with support on E and dim(E\cap l)=\beta(\theta) for Leb-almost all l\in L_{\theta}.

The non rational case was recently studied by Pablo.

Theorem [P. Schmerkin, 2016] Let E\subset \mathbb{R}^2 a closed set and invariant for T_p(x,y)=(T_p(x),T_p(y)). If \tan \theta \notin \mathbb{Q} then dim(E\cap l)\leq \max(dim(E)-1,0) for all l\in L_{\theta}.

The proof of the theorems follows from an application of a tool now known as the BSG (Balog-Szemer\’edi-Gowers) theorem, which has found many further applications. In the words I can remember… is a theorem in combinatorics proved with graphs that allows to have a structure on the integeres, it allows to find lower bounds by convoluting indicatrix functions and obtained a l^2 norm that can by bound with an l^1 norm apart from a small set. Remember that the BSG theorem is the main ingredient of the 1998 proof of the first effective bounds for the Szemer\’edi theorem, showing that any subset A\subset\{1,2,\ldots,N\} free of k-term arithmetic progression has cardinality O(N(\log \log N)^{-c_k}) for an appropriate c_k>0.

Notice that the theorem works in particular for E=C\times C. Pablo was not convinced of the conjecture that indeed dim(E\cap l)\in \{dim(E)-1,0\}.

As a corollary we conclude that if \lambda\notin\mathbb{Q}, then for all t\in \mathbb{R}, dim(C\cap (\lambda C +t))\leq 2\frac{\log 2}{\log 3}-1.

The following theorem was obtained independently in 2016 by P. Schmerkin and M. Wu:

Theorem [P. Schmerkin, 2016] If p and q are not powers of the same integer, A,B are closed sets, T_p,T_q-invariants, then
dim(A\cap (\lambda B+t))\leq \max(dim(A)+dim(B)-1,0) for all \lambda\neq 0, t\in\mathbb{R}.


A random meeting with a formal ergodic theorist

Most of the people, say more than 99.999999% of the population, has no idea of what Ergodic theory is. Therefore, if you meet someone randomly, you may be sure that he/she does not have idea at all of what it does mean. Among many other examples you can tell in order to introduce the concept of ergodicity, today,  Jana Rodríguez Hertz told us one that was quite nice and that I would not like to keep to myself.

The example Jana told us was the following:

Suppose you want to know the number of fish in a lake, say n. It is extremely difficult to count them, however you can estimate n. In order to do this, you can take p fish and mark them, then allow them to return to the lake. After some time, you take m fish and you count how many are marked. If  you assume that after enough time the proportion of marked fish is the same at “every scale”, i.e., if you take any amount k of fish you will always obtain a number r(k) of marked ones so that \frac{r(k)}{k}=\frac{p}{n}, then you can estimate n by \frac{p m}{r(m)}, where r(m) is the number of marked fish you obtained when you took in total m fish. Mathematically the validity of this statement is justified if you have the mixing property. This is a strong condition of the system.

Another strategy for estimating n is to capture m fish each 10 days, count how many are marked, and then allow them to return to the lake. Suppose the i-th time you captured fish you obtained r_i marked ones. If you do this many times, say N, then you can estimate n by \frac{Np}{\sum_{i=1}^N\frac{r_i}{m}}= \frac{Npm}{\sum_{i=1}^N r_i}.  Mathematically the validity of this statement is justified by the ergodic property. This is weaker than the mixing condition of the system needed in the previous estimation strategy.

In formal terms you may introduce the concept of ergodicity as the behavior of a system where the second estimation approach does work. Sadly or luckily, who knows, ergodicity is unlikely to occur in most of the dynamical systems in nature :).