Schmidt Games

In this lecture we will discuss the definitions of (alpha,beta)-Schimdt games, (alpha,beta)-winning sets and alpha-winning sets. We will see some examples of games and prove that every countable intersection of alpha-winning sets is itself also an alpha-winning set. Finally, we will discuss optimal strategies in the game and show the existence of "position" strategies that are optimal. The lecture is based on the following paper of Schmidt. Here are the notes from Itamar's lecture.

Badly Approximable Vectors on Fractals

Itamar will prove the countable intersection theorem stated last time.

In Oded's talk we will use Schmidt games to show the existence of real vectors that are badly approximable by rational vectors, in any dimension. The method we'll discuss also extends the result to the case of vectors that lie in a nice enough fractal, and shows that such fractals contain points whose images under any given countable family of affine transformations are all badly approximable.

The talk is based on the paper of Schmidt discussed last week, and on this paper of Fishman. Here are the notes for the talk.

Hausdorff dimension of winning sets

We will show that every winning set in R^n has full Hausdorff measure, but in general in complete metric spaces this is not the case. The talk is based on sections 2.4 and 5.1 of this paper. Here are the notes for Ofek's lecture.

Based on the paper of Gale and Stewart, Infinite games with perfect information, in Contributions to the theory of games, p. 245-266, Annals of Mathematics Studies no. 28, 1953. Here are the notes of the lecture.

Following the paper The Set of Badly Approximable Vectors is Strongly C^1 incompressible by Broderick et al, we introduce a new version of Schmidt games that involves intersection with hyperplanes. Using the winning sets in these game types (called `hyperplane absolute winning', or HAW) we prove that the set of Badly approximate vectors is strongly C^1 incompressible. This property is essentially a lower bound on the Hausdorff dimension of the intersection with C^1 images of the set with itself. Finally, we state some results regarding games played inside fractals in R^n.

Here are the notes for the lecture.

Borel hyperplane absolute winning sets in R^d

We discuss two equivalent formulations of variations of Schmidt games, namely hyperplane absolute winning game (HAW) and restricted hyperplane absolute game. We will define a class of sets in R^d known as hyperplane diffuse (introduced in [BFK+12]). Following [BNY21] we will show that if a set S in R^d is HAW then its intersection with any hyperplane diffuse set is not empty. Moreover, if S is Borel then the converse also holds. To prove this we will discuss a Borel determinacy theorem for absolute winning games appearing in [FLS14].

[BNY21] Beresnevich, V., Nesharim, E. & Yang, L. Bad(w) is hyperplane absolute winning. Geom. Funct. Anal. 31, 1–33 (2021). Bad(w) is hyperplane absolute winning. Geom. Funct. Anal. 31, 1-33 (2021).

[BFK+12] Ryan Broderick, Lior Fishman, Dmitry Kleinbock, Asaf Reich, and Barak Weiss. The set of badly approximable vectors is strongly C1 incompressible. Math. Proc. Cambridge Philos. Soc., 153(02):319-339, 2012.

[FLS14] Lior Fishman, Tue Ly, and David Simmons. Determinacy and indeterminacy of games played on complete metric spaces. Bull. Aust. Math. Soc., 90:339-351, 2014 Here are the notes for the lecture.

14:10, PLEASE NOTE SPECIAL TIME

Crash course on the geometry of Lattices

We will talk about Lattices in R^n and their geometrical properties -- basic definitions and results, Minkowski's successive minima and the 1st & 2nd theorems, j-dimensional sub-lattices and the alpha_j parameters, the Harder-Narasimhan filtration. Talk will be roughly based on the Geometry of Numbers course-notes given by Barak Weiss last fall, lectures 1-5.

Here are the notes for the lecture.

Introduction to homogeneous dynamics

In this lecture, we will discuss some basic notions and prove fundamental results, in the theory of homogeneous dynamics. Our main focus will be on the action of the special linear group on the space of lattices in R^n, and we will develop dynamical tools that will later help us in the study of Diophantine approximations. We will discuss in depth Mahler's compactness criterion and Dani's correspondence theorem.

Here are the notes for the lecture.

Overview of the dimension theory of singular vectors and matrices

We will construct some examples of singular vectors in the plane, and show they have measure zero. Then we will discuss the incremental progress made over the last decade toward calculating the dimension of singular vectors and matrices, which culminates in Das-Fishman-Simmons-Urbanski's recent work. They establish a "variational principle" for a class of functions called "templates," which formalize the properties of the Minkowski successive minima of the orbits of certain unimodular lattices under the diagonal flow given by the Dani correspondence principle. This principle will be studied in depth next semester, but in this talk we will see how it can be used to calculate the dimension of singular vectors and matrices.

Here are the notes for the lecture.

The dimension formula for singular vectors

The new variational principle in the parametric geometry of numbers reduces the computation of the packing and Hausdorff dimension of singular matrices to the study of combinatorics of templates. In the case of singular vectors, we will show how these combinatorial arguments give some of the desired dimension formulas.

Selected applications of the variational principle

Two applications of the variational principle will be presented, due to [DFSU]: Theorem 3.12 (motivated by a conjecture of Starkov, solved by Cheung) computes the dimension of vectors simultaneously singular, and not very well approximable. Theorem 3.14 (motivated by a conjecture of Schmidt, solved by Moshchevitin) computes the dimension of lattices whose k-1-st successive minimum goes to zero, but k+1-st successive minimum goes to infinity.

Here are the notes for the lecture.

Back to Schmidt Games: Hausdorff dimension and winning sets

We will define a new variant of Schmidt games, and prove that the Hausdorff and packing dimensions of Borel sets can be calculated using this variant game. If time permits, we will also present a modified version of this variant, that will be useful for the proof of the "variational principle". This lecture is based on section 3 of [DFSU].

Here are the notes from Itamar's lecture.

Overview of proof of variational principle, and reduction

The variational principle in the parametric geometry of numbers established by Das, Fishman, Simmons, and Urbanski [DFSU] describes the Hausdorff dimension of matrices whose associated lattice (via the Dani correspondence) obeys a given template (or collection of templates). In this lecture, we will first provide a rough outline of the proof with a focus on the lower bound on the Hausdorff dimension. After that, we will begin the proof by establishing a reduction to certain simpler templates. In particular, the corner points of these simplified templates lie in a mesh. The lecture is based on Section 31 and 32 in [DFSU]. Here is the handout.

Overview of proof of variational principle, and reduction (continuation)

Mini strategy

We will present the first part of the proof of the variational principle from the DFSU paper. We will construct the "mini-strategy", which is a local strategy for Alice in the Hausdorff game. The strategy will be our main tool for proving the variational principle later on in the seminar. In the lecture, we will present Alice's strategy and show that it is efficient. The lecture will be based on section 31 in the DFSU paper.

Mini strategy (continued)

Error correction

The global strategy for the Hausdorff game consists of repeated use of the mini-strategy defined in the previous lecture. However, the mini-strategy involves an error term that compounds with this repetition, which can spoil the desired estimates. To control this error we will introduce `perturbation vectors' and show how they can be used, together with the mini-strategy, to achieve the desired lower bounds in the proof of the variational principle.

Here are the notes for the lecture.

Error correction (cont.)

Yom Ha-atzmaut (no meeting)

14:10, room 209 PLEASE NOTE SPECIAL TIME AND PLACE

Uniform error correction

Following Section 31.4 in DFSU paper, we describe the global strategy of Alice that guarantees lower bounds on the Hausdorff and packing dimensions of the set D(f). We use perturbation vectors to define the strategy. The applicability of the strategy then follows from boundedness properties of these vectors. We prove that this is indeed the case, namely that the perturbation vectors are uniformly bounded.

Uniform error correction (continued)

14:10 PLEASE NOTE SPECIAL TIME

The upper bound

The last part in the DFSU paper completes the proof of the variational principle. We will prove the upper bound for the Hausdorff and packing dimensions of the set D(S). This time, we will define a strategy for Bob for sufficiently small beta, which guarantees a bounded score for Alice, which in turn bounds the Hausdorff (packing) dimension of the target set from above.

PLEASE NOTE SPECIAL TIMES

Parametric geometry of numbers: the weighted case.

We will cover the changes we can perform to DFSU to adapt it to the weighted setting. We will introduce:

New templates: In order to correctly define the weighted template we will investigate dynamics of action of the diagonal group on the space of linear subspaces of ℝn.

New dimension: We will change the metric in which we compute the Hausdorff dimension. We will then discuss the main geometric theorem which gives intuition to the dimension formula.

Template perturbation: we will employ the new definition of template and see a different and more local way to perturb templates.

The strategy of Alice: we will discuss how to define the strategy without using the conditions.