On x2 x3 and equidistribution by Host's theorem.

In his seminal disjointness paper, H. Furstenberg proved that for every irrational number x, the orbit of x under the action of the semigroup generated by two multiplicatively independent numbers in the torus is dense. Later on, Furstenberg raised the question about classifications of measures which are ergodic+invariant under such semigroups, the question was partially answered by D. Rudolph and A. Johnson. I will present a different approach to the subject, based on a theorem of B. Host (and D. Meiri) which can be thought of as equidistribution result about fractal measures. If time permits, I will also discuss recent improvements of Rudloph's theorem done by Bourgain-Lindenstrauss-Michel-Venkatesh which gives a quantified version of Furstenberg's result.

Based on the paper Nombres normaux, entropie, translations by Bernard Host, Israel Journal of Mathematics, 1995, Volume 91 419-428.

Exercises.

1. Prove that 3 is invertible modulo 2, and in general modulo 2^n for any n.

2. Calculate the size of <3> modulo 2^n, or at-least give a meaningful lower bound for it.

2.5 For an extra credit of sizable epsilon points, one can think/numerically experiment about the size of sets such as {3^{3^n}} mod 2^m as m>>0.

3. For the usual Cantor set K, show directly that <2>*K is dense modulo 1.

4. [For those who know entropy theory] Show that for any positive h < log p, there exists an ergodic measure supported in the torus of entropy h, which is xp invariant (and in particular, its support is a closed invariant subset different from the whole torus).

5. Is it true that for any irrational number x the set <2,3>x is equidistributed in the torus? [Hint - think about Liouvillian numbers of basis 6].

The limiting distribution of random Cayley graphs

Given a random Cayley graph we wish to study the limiting distribution of its diameter, which is the maximal distance between any pair of vertices. Marklof and Strombersson used the limit distribution for Frobenius numbers in m+1 variables to prove that the diameter of a random Cayley graphs of Z/kZ with a generating set of fixed size m>1 has a limit in distribution and found that limit. In this talk we survey their result and expand the discussion to Cayley graphs of Z^n/Sigma, with a generating set of fixed size m>n, where Sigma is a sublattice of Z^n.

Intrinsic Diophantine approximation on quadratic hypersurfaces

We consider the question of how well points in a quadric hypersurface $M\subset\mathbb R^d$ can be approximated by rational points of $\mathbb Q^d\cap M$. This question turns out to be closely related to the dynamics of a certain flow on the homogeneous space $G/\Gamma$, where $G$ is the group of matrices preserving a certain quadratic form on $\mathbb R^{d + 1}$ and $\Gamma$ is the group of integer matrices in $G$. This connection is an analogue of the so-called Dani--Kleinbock--Margulis correspondence principle relating approximation of points in $\mathbb R^d$ by points in $\mathbb Q^d$ to the dynamics of a certain flow on the space of unimodular lattices in $\mathbb R^{d + 1}$. This work is joint with Lior Fishman, Dmitry Kleinbock, and Keith Merrill.

One-dimensional Substitution Tilings with an Interval Projection Structure (following Harriss and Lamb).

The talk will discuss this paper . We study nonperiodic tilings of the line obtained by a projection method with an interval projection structure. We obtain a geometric characterisation of all interval projection tilings that admit substitution rules and describe the set of substitution rules for each such a tiling. We show that each substitution tiling admits a countably infinite number of nonequivalent substitution rules. We also provide a complete description of all tilings of the line and half line with an interval projection structure that are fixed by a substitution rule. Finally, we discuss how our results relate to renormalization properties of interval exchange transformations (with two or three intervals).

x2, x3-invariant measures on the torus (following Rudolph)

In his seminal disjointness paper, H. Furstenberg proved that for every irrational number x, the orbit of x under the action of the semigroup generated by two multiplicatively independent numbers in the torus is dense. Later on, Furstenberg raised the question about classification of measures which are ergodic and invariant under such semigroups. The question was partially answered by D. Rudolph (x2 and x3 invariant measures and entropy, ETDS 10, 1990) assuming that both numbers are co-prime and one on the transformation has positive entropy. We will try to outline his proof following this exposition.

Here are some exercises.

Diophantine Estimates in Positive Characteristic

We study the Diophantine Approximation problem for function fields of positive characteristic.We begin with a brief overview of developments in the theory of Diophantine approximation. Using techniques developed by Athreya, Parrish and Tseng we prove a weaker version of Schmidt's theorem, on distribution of solutions for the Diophantine problem, for function fields of positive characteristic.

Quantitative Recurrence (following Boshernitzan)

The Poincare recurrence theorem is a fundamental result which states that any measure-preserving system exhibits a nontrivial recurrence to any measurable set with positive measure. In the case of metric spaces, the theorem can be stated as follows: let (X,d) be a separable metric space, T: X -> X a transformation and mu a T-invariant Borel probability measure, then for almost all x, liminf_n d(x, T^nx) =0.

We will see that under some additional assumptions the rate of recurrence can be quantified and it depends on the Hausdorff dimension of X. We will also see some applications of the theorem. Based on the paper M. Boshernitzan, Quantitative recurrence results, Inv. Math. 1993.

Attached are some exercises .

Peres - Schlag probabilistic method in Diophantine approximation.

The abstract and exercises are here.

13:10, please note special time!

Schmidt's games and applications.

In my talk I will describe Schmidt's original game as well as some more recent variants of this game. I will mention some properties of winning sets and talk about applications of these results to Diophantine approximations and fractals. The lecture will be based mainly on the following paper:

"THE SET OF BADLY APPROXIMABLE VECTORS IS STRONGLY C1 INCOMPRESSIBLE" by RYAN BRODERICK, LIOR FISHMAN, DMITRY KLEINBOCK, ASAF REICH AND BARAK WEISS (available here) Here are some exercises.