Organizational meeting

Niclas Technau

A survey of results and questions of Diophantine approximation

Niclas Technau and Florent Ygouf

Diophantine approximation on manifolds, and Dani correspondence.

Niclas will describe some well-known results in Diophantine approximation on manifolds. They are contained in the following three papers: paper of Kleinbock and Margulis on extremality of manifolds, paper of Beresnevich on badly approximable vectors on manifolds, and paper of Shah on Dirichlet improvable vectors on manifolds. Here are the notes to Niclas' talk.

The Diophantine approximation concepts introduced by Niclas can be interpreted in term of divergence of flows on the space of unimodular lattices. Florent will explain this interpretation and give a dynamical criterion for each the concepts presented by Niclas. This correspondence usually goes under the name of Dani correspondence. Here are the notes for Florent's talk.

Some references for Florent's talk: This survey paper by Dmitry Kleinbock introducing the history and context, but not with full details, this 1985 paper of Dani in which the Dani correspondence first appeared in connection with singular and badly approximable vectors and matrices, this 1998 paper of Kleinbock and Margulis establishing logarithm laws and making the connection with psi-approximability more precise, and this 2008 paper of Kleinbock and Weiss about Dirichlet improvability.

More Dani correspondence, and continued fractions.

Florent will continue with his presentation, giving the homogeneous interpretation of notions of multiplicative approximation and weighted approximation. Here are the notes for Florent's talk.

After that, Manuel will describe the relationship between continued fraction expansions and dynamics of the diagonal flow on SL(2,R)/SL(2,Z). He will present the results of section 13 of [SW].

Geodesic flow, and equidistribution with respect to the Gauss map.

We introduce the continued fraction expansion of an (irrational) real number, the Gauss map, and in particular the Gauss measure, which is a shift invariant, ergodic probability measure on the space of all continued fraction expansions in the unit interval. In particular, the orbit of almost every continued fraction expansion equidistributes with respect to the Gauss measure. Following this, we deduce a sufficient condition for an individual point to equidistribute. More precisely, we show that equidistribution under the geodesic flow of a point on the standard horocycle in the modular surface implies equidistribution of the corresponding continued fraction expansion under the shift map.

Here are the notes for Manuel's talk.

Bootstrapping from equidistribution on a small space, to equidistribution on a larger space.

In this lecture, we will mainly follow section 5 of [SW]. The main purpose of this section is to upgrade the second conclusion of Theorem 2.1 of [SW] to Theorem 2.2. In particular, we will prove Propositions 5.1 and 5.2 and state Proposition 5.3 and Corollary 5.5. The proofs will require some background on probability theory (conditional expectations, martingale convergence theorem) which we will review in the lecture. The main references for this probability background are Chapter 5 of the book "Ergodic theory with a view towards number theory", by Einsiedler and Ward, and appendix A of the book "Random walks in reductive groups", by Benoist and Quint.

Here are notes for Shucheng's lectures.

Bootstrapping (cont.), and Deriving the Diophantine Properties of IFS Fractals

Shucheng will continue last week's talk, see above for the abstract. Itamar will start discussing the connection between random walks and fractals. He will introduce the concepts if IFSes - Iterated Function Systems and the fractals they define. In addition he will introduce the concept of random walks on the space of lattices. Using a theorem regarding equidistribution of the random walk trajectory for such cases (that we shall prove later on in the seminar) and machinery from the previous meetings he will prove Diophantine results for these types of fractals.

Here are notes for Itamar's lectures.

Deriving the Diophantine Properties of IFS Fractals (cont.)

Itamar will continue last week's lecture, see notes above.

From classification of stationary measures to description of trajectories

The main theorem of [SW] says that for a certain type of random walk: (i) any stationary measure is induced from a Haar measure, and (ii) the random walk is equidistributed w.r.t the Haar-induced measure.

We will show that (i) implies (ii), using ideas from the three Benoist-Quint papers, the work of Eskin-Margulis, and Breiman law of large numbers.

Sources: Benoist-Quint book Random walks on reductive groups , 2.3+2.4 and 1.7+1.8 from the appendix.

Benoist-Quint I, 6.3 (click here for the English translation. )

Benoist-Quint II 6.3

Benoist-Quint III 3.5+3.7+3.9+3.10

Eskin-Margulis 3.1+3.3+3.4+4.2

From classification of stationary measures to description of trajectories (cont.)

Tsviqa will complete last week's lecture.

Oseledec theorem

The lecture will present the statement and proof of the theorem of Oseledec. The proof presented is the following proof by Raghunathan based on some results of Furstenberg and Kesten.