I will write about an interesting talk given by Pablo Shmerkin.
Main goal: computing for with dynamics properties.
Theorem [Marstrand, 1954] be a Borelian set. Given we define to be the family of all the lines on the plane with slope Then for almost all we have that
The graph of a Brownian motion on the plane has however, vertical lines have intersection of HDIM equal to zero.
Theorem [Marstrand, 1954] Borelian set.
We can obtain the following corollaries for the HDIM of the intersection of subsets
Corollary Let be Borelian set. Then for Leb almost all we have that
Remember that and that there is equality if Therefore, we have the following corollaries.
Corollary If is a Borelian set such that then for Leb almost all we have that
And from the second theorem of Marstrand, we have the following corollary.
Corollary where is lineal.
There exists compact such that the cardinality of such that and
For all and for all [Falconer example]
Furstenberg asked what happened if were invariant for the map Motivated by this questions, let us consider first the classic Cantor set whose Applying directly Marstrand’s theorem we obtain the bound for almost all There is however a better bound by Hawkes.
Theorem [Hawkes, 1975] for almost all
Kenyon and Peres generalized further in 1991 the result to the case of Notice that
Theorem [R. Kenyon and Y. Peres, 1991] For almost all Where greater Lyapunov exponent for the product of iid matrices where
Barany and Rams in 2014 considered the case of self-similar sets that are obtained by a generalization of the set (ex. consider the unit square divide it in 9 squares of side numerate the squares by . Choose squares, say delete the rest of the squares and repeat the procedure on each of the squares left, etc.).
Theorem [B. Barany and M. Rams, 2014] Let a set constructed doing the described process where and If then with for -almost all where is a natural measure with support on and for Leb-almost all
The non rational case was recently studied by Pablo.
Theorem [P. Schmerkin, 2016] Let a closed set and invariant for If then for all
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 norm that can by bound with an 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 free of -term arithmetic progression has cardinality for an appropriate
Notice that the theorem works in particular for Pablo was not convinced of the conjecture that indeed .
As a corollary we conclude that if then for all
The following theorem was obtained independently in 2016 by P. Schmerkin and M. Wu:
Theorem [P. Schmerkin, 2016] If and are not powers of the same integer, are closed sets, -invariants, then