We will discuss the main results of the paper, which were introduced briefly in our first seminar meeting. Specifically, for a norm on R^d, a vector theta, and positive epsilon, will talk about the statistical behavior of the sequence v_k (best approximations) and the sequence w_k (epsilon -approximations), focusing on their length, direction, and arithmetical properties. We will then discuss the statistical behavior of the "projected lattices" which correspond to these v_k and w_k sequences. If time permits, will also discuss a special non-typical case, where the coordinates of theta form a basis, with 1, to a totally real number field (in this case theta is badly-approximable). Here are the notes for both of Alon's lectures.

Alon will complete his lecture by discussing another theorem from the introduction, about the behavior of vectors with coefficients in a totally real number field of degree d+1. Vika will motivate and (time permitting) state a more refined result in section 2 of the paper, about the "bundle of lift functionals". The discussion will be based on sections 1,2 and subsection 3.2 of the paper.

Here are the notes for Vika's talk.

Vika will continue her talk from last week, based on section 2 of the paper. Rishi's talk explores how the study of continuous-time dynamical systems can be simplified by examining the discrete-time systems of the first return map. He will discuss Kac's Lemma, Kakutani Towers, flows under a function, cross-sections, and Theorem 4.4 of the paper. If time permits, he will also cover the material in Section 4.2.

Besides the paper, the lecture is based on Chapters 1 and 7 of the book Basic ergodic theory by Nadkarni, and Section 2.9 of the book Ergodic theory with a view toward number theory by Einsiedler and Ward. Here are the notes for the talk.

Here are the notes for Rishi's second talk.

We introduce the notion of a reasonable cross section for a flow on a (sufficiently nice) topological probability space and explore the relationship between generic points of the flow and points whose sequence of visits to the cross section equidistributes. Here are the notes for Rotem's talk.

We will look at Chapter 6, where we lift cross sections along factor maps, and study its properties. If the factor map is furthermore a fiber bundle map, then a lift of a reasonable cross section (as in chapter 5) is also reasonable.

Here are the notes for notes for Alon's second talk.

Note special day and time

Here are the notes for Yuval's talk.

Here are the notes for Chen's talk.

The talk will discuss the properties of the cross section measures and their projections on various factors. Only Case I will be discussed, and the adeles will be avoided. Based on section 11.1 of the paper. Here are the notes for Vika's talk.

Alon Agin, Temperedness and lack thereof in Case I

The talk will focus on the temperedness (or lack of temperedness) for the cross-sections for best and epsilon approximations, and how to overcome the issue of non-temperedness. Most of the material is in sections 9 and 12. It will be good to remind yourselves of the contents of Rotem's lecture on Jan. 2.