Dirichlet’s approximation theorem states that for every real number x there exist infinitely many rationals p/q with |x-p/q| < 1/q^2. If x is in the unit interval, then viewing rationals as algebraic numbers of degree 1, q is also the height of its primitive integer polynomial, where height means the maximum of the absolute values of the coefficients. This suggests a more general question: How well can real numbers be approximated by algebraic numbers of degree at most n, relative to their heights? We will discuss Wirsing’s conjecture which proposes an answer to this question and Schmidt and Davenport’s proof of the n = 2 case, as well as some open questions.

We show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions that exceeds 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of homogeneous Cantor sets). We will also discuss the connection of this problem with the question on properties of one dimensional self-similar sets with overlaps.

Consider a random product of i.i.d. matrices, randomly chosen from SL(2,R), satisfying some reasonable nondegeneracy conditions (no finite common invariant set of lines, no common invariant metric).Then a classical Furstenberg theorem implies that the norm of such a random product almost surely grows exponentially.

Now, what happens if each of these matrices depends on an additional parameter? We will discuss the case when the dependence of angle is monotonous w.r.t. the parameter: increasing the parameter «rotates all the directions clockwise».

It turns out that (under some reasonable conditions)
- Almost surely for all the parameter values, except for a zero Hausdorff dimension (random) set, the Lyapunov exponent exists and equals to the Furstenberg one.
- Almost surely for all the parameter values the upper Lyapunov exponent equals to the Furstenberg one
- At the same time, in the no-uniform-hyperbolicity parameter region there exists a dense subset of parameters, for each of which the lower Lyapunov exponent takes any fixed value between 0 and the Furstenberg exponent.

The talk is based on a joint project with A. Gorodetski.

My talk will be related to the stable intersections of dynamically defined Cantor sets (as well as to the finding intervals in their sums). In the famous work of Morreira and Yoccoz it is shown that for a generic pair of dynamically defined Cantor sets with sum of their Hausdorff dimensions greater than one, their intersection (provided that they do intersect) is stable under small perturbations.

However, it would be nice to be able to check this for a particular couple of sets, and up to this moment the only explicitly checkable sufficient condition that is known is that the product of the thicknesses is greater than one.

I will present an approach to finding such a sufficient condition (eventually, in a computer-assisted way) that can be hopefully used to attack the conjecture that claims that F(2)+F(4) contains an interval (that is currently open).

My talk will be based on a joint project will A. Gorodetski, A. Gordenko and E. Nesterova.

We will talk about a paper by A. Moreira and J.C. Yoccoz, where they proved a conjecture by Palis according to which the arithmetic sums of generic pairs of regular Cantor sets on the line either has zero Lebesgue measure or contains an interval.