Chebotarev's result on Minkowski's conjecture

following Woods.

One of the best results on Minkowski's conjecture (a longstanding open problem in geometry of numbers) is a result of Chebotarev from 1934. I will recall Minkowski's conjecture and explain the proof following a presentation of Woods. The proof involves (implicitly) some dynamical ideas concerning the action of the diagonal group on the space of unimodular grids. I will try to make these ideas more explicit.

Here is the relevant paper.

Siegel Measures (following Veech)

Consider the space $M$ of all locally finite Borel measures $\nu$ on euclidean n-space satisfying $\nu(B(R))=O(R^n)$, where $B(R)$ denotes the ball of radius $R$. There is a natural action of $G=SL_n(R)$ on $M$. Call a $G$-ergodic probability measure $\mu$ on $M$ a Siegel measure. For $f:\R^n\to \R_+$ define the functional $f^\hat(\nu)=\nu(f)$, then $\mu(f^\hat)=c m(f)$ where $m$ is the Lebesgue measure and $c$ a constant depending on $\mu$. It is a generalisation of a classical identity due to Siegel (1944), which has been used to calculate the volume of $SL_n(R)/SL_n(Z)$ and give a proof of Minkowski-Hlawka theorem on sphere packing. Veech used this formula to give a mean asymptotic for counting problems on translation surfaces. In this talk, we restrict to the abstract setting following Veech (1998).

Suggested reading: it will be instructive to read the introduction of the paper of Veech, to get used to the abstract setup.

Delone Sets and their Inflation Symmetries (following Lagarias).

A natural way to classify Delone sets X which have some sort of weak translational order is by imposing increasingly strong restrictions on their sets of interpoint vectors X - X. We call sets for which the abelian group [X - X] is finitely generated, finitely generated Delone sets, sets for which X - X is locally finite, Delone sets of finite type and sets for which X - X is itself Delone are called Meyer sets. We say a Delone set X has inflation symmetry if there exists a real number a>1 such that aX is a subset of X. Following Lagarias and Meyer we will discuss some properties of Delone sets of the three classes defined above and study the algebraic properties of inflation symmetry constants associated with Delone sets of these classes which have inflation symmetries.

Reading: Jeff Lagarias - Geometric Models for Quasicrystals I. Delone Sets of Finite Type. Discrete & Computational Geometry 21 (1999), pp. 161-191. Available on Lagarias' webpage.

HW: Get better acquainted with the definitions and perhaps read proof that Meyer \subset of finite type \subset finitely generated. I might use but won't prove the main results of ch. 2 and 3 of the paper, which give equivalent conditions for being of finite type and Meyer respectively.

Distribution of Packets of Periodic Torus Orbits

Following Einsiedler, Lindenstrauss, Michel and Venkatesh (ELMV09, ELMV11) we will first discuss the properties of some finite collections of periodic orbits for the diagonal subgroup of PGLn(R) acting on the space of lattices, PGLn(Z)\PGLn(R). These arithmetic collections are called packets. Packets of periodic diagonal orbits are closely related to class groups of orders in totally real number fields of degree n. We will not assume familiarity with algebraic number theory.

In the second part we will talk about bounds on the asymptotic entropy of packets with volume going to infinity. We will discuss some of the following methods for bounding the entropy: Linnik's method in rank 1 (ELMV12), a bound based on the adjoint representation (ELMV09), a bound based on the double quotient of PGLn by a torus (Kh15) and a bound on the entropy of all ergodic components using a period formula for the Eisenstein series and subconvexity (ELMV11). I will try to show how ELMV09 and Kh15 generalise various aspects of Linnik's work.

References:

ELMV12: A modern variant of Linnik's and Skubenko's original argument in rank 1 adapted to entropy. M. Einsiedler, E. Lindenstrauss, P. Michel and A. Venkatesh. The distribution of closed geodesics on the modular surface, and Duke's theorem. Enseign. Math. (2) 58 (2012), no. 3-4, 249--313.

ELMV09: Definition of packets and their discriminant for PGLn(R) in local terms, a bound on the entropy using the adjoint representation. M. Einsiedler, E. Lindenstrauss, P. Michel and A. Venkatesh. Distribution of periodic torus orbits on homogeneous spaces. Duke Math. J. 148 (2009), no. 1, 119--174.

Kh15: A bound on the entropy for some packets using the double torus quotient and the action of the Galois group of a splitting field. I. Khayutin. Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbits.

ELMV11: Definition of packets in adelic terms, a bound on the entropy of every ergodic component using Eisenstein series and subconvexity. M. Einsiedler, E. Lindenstrauss, P. Michel and A. Venkatesh. Distribution of periodic torus orbits and Duke's theorem for cubic fields. Ann. of Math. (2) 173 (2011), no. 2, 815--885.

EMV13: The following paper includes an in depth discussion of Linnik's original argument which used the theory of large deviations rather than entropy. J. Ellenberg, P. Michel and A. Venkatesh Linnik's ergodic method and the distribution of integer points on spheres.

Automorphic representations and $L$-functions, 119--185, Tata Inst. Fundam. Res. Stud. Math., 22, Tata Inst. Fund. Res., Mumbai, 2013.

Divergent trajectories for the diagonal group and split tori (following Margulis, Tomanov and Weiss)

We begin with discussing divergent trajectories for the case of diagonal group in special linear groups. This is a special case of divergent trajectories for a maximal split torus in an algebraic group. For more general results we recall some definitions and properties concerning algebraic group. In this context, a new compactness criterion will be defined using horospherical subsets.

Reading: Preliminaries section (2) in the paper - G. Tomanov, B. Weiss, Closed orbits for actions of maximal tori on homogeneous spaces , Duke Math. J. 119 (2003), no. 2, 367-392.

HW problem: If rho: G -> GL(V) is defined over Q and there is v in V(Q) such that rho(x_n)v tends to 0 as n -> infinity, then x_n\Gamma tends to infinity in G/G(Z).

Entropy and dimension for shift invariant subsets (following Furstenberg)

We recall the definitions of Minkowski and Hausdorff dimension in metric spaces, and define the topological entropy of subsets of compact dynamical systems. We will focus on the case of symbolic dynamics over a finite alphabet. Following the paper "Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation" of Furstenburg from 1967, we show that for shift invariant subsets, Hausdorff dimension = Minkowski dimension = Entropy / log (size of the alphabet). We mention the analogous result for invariant measures, which is closely related to the Shannon-McMillan-Breiman theorem.

Furstenberg's paper is here, and here is the exercise sheet.

Cut and project sets and Diophantine approximation

Cut and project sets give a way of defining discrete point patterns through a data of a linear subspace and an acceptance strip. The points in the pattern are obtained through first intersecting the integer lattice with the acceptance strip (cut) and then projecting these intersection points to the subspace (project). We establish a connection between finite patches in cut and project sets and an action of a toral rotation defined by the cut and project data.

Based on this connection, we use methods from Diophantine approximation to measure how well ordered cut and project sets are, in terms of repetitivity functions and pattern frequencies. Finally, we see that the existence of extremely well ordered cut and project sets turns out to be equivalent to the negation of the Littlewood conjecture.

The talk is based on several works, joint with Alan Haynes, Antoine Julien, Lorenzo Sadun and James Walton.

Continuum biLipschitz classes of Delone sets in R^d

I will sketch a proof that all Delone sets in R^d (d \geq 2) are split into continuum many classes under biLipschitz equivalence.

The work was inspired by the remarkable result of D.Burago and B.Kleiner, and, independently, C. McMullen (1998). They proved the existence of Delone sets in R^d that are not biLipschitz equivalent to the lattice Z^d.

For the lower bound "at least continuum" we will use Burago and Kleiner's technique to produce some local patterns that we will later use as patches to Z^d. Then we will show continuum many ways to place these patches in R^d so that the resulting sets are pairwise not biLipschitz equivalent.

Ex. Every Delone set is biLipschitz equivalent to some subset of Z^d.

The upper bound "at most continuum" follows from this fact (cf. Garber, "On Equivalence Classes of Separated Nets," Model. Anal. Inf. Syst. 16 (2), 109 2010).

On quantitative disjointness in unipotent dynamics.

In 1941 Wiener-Wintner proved their famous theorem about convergence of twisted ergodic averages, a result that nowadays can be considered as an instance of disjointness of weak-mixing systems from Kronecker systems. In the talk I will discuss a recent result due to Venkatesh which proves a quantified version of the theorem for the horocyclic flow on cocompact surfaces and illustrate some nice applications. The proof follows Bourgain's proof of the uniform Wiener-Wintner theorem, combined with Furstenberg's equidistribution theorem for the horocyclic flow on compact homogeneous spaces and bounds for matrix coefficients.

The talk will be accessible, I will start by proving Bourgain's uniform Wiener-Wintner theorem, deduce a qualitative version of Venkatesh's result and as time permits, explain the quantification and in particular, some bounds for matrix coefficients.

Some recommended preliminary reading: 1. The chapter about joinings in the book of Manfred Einsiedler and Tom Ward, or even better in Rudolph's book.

2. For quantitative mixing, we would need some non-abeliean harmonic analysis, I've tried to find reference from the lightest weight to the heaviest weight.

2.1 The introduction article - SPECTRAL THEORY OF AUTOMORPHIC FORMS: A VERY BRIEF INTRODUCTION by Venkatesh.

2.2 Automorphic forms on SL2 by Borel.

2.3 Knapp-Trappa - Representations of semisimple Lie groups , lecture 7, and Knapp's book.

Self-affine measures with equal Hausdorff and Lyapunov dimensions

Let $\mu$ be the stationary measure on $\mathbb{R}^{d}$ which corresponds to a self-affine iterated function system $\Phi$ and a probability vector $p$. Denote by $\mathbf{A}\subset Gl(d,\mathbb{R})$ the linear parts of $\Phi$. Assuming the members of $\mathbf{A}$ contract by more than $\frac{1}{2}$, it follows from a result by Jordan, Pollicott and Simon, that if the translations of $\Phi$ are drawn according to the Lebesgue measure, then $\dim_{H}\mu=\min\{D,d\}$ almost surely. Here $D$ is the Lyapunov dimension, which is an explicit constant defined in terms of $\mathbf{A}$ and $p$.

I will present a new result which provides general conditions for $\mu$ to be exact dimensional with $\dim\mu=D$, whenever $\Phi$ satisfies strong separation. These conditions involve a lower bound on the dimension of the Furstenberg measure corresponding to $\mathbf{A}$ and $p$. The proof uses random matrix theory, and upper bounds on the dimension of exceptional sets of sections and projections of measures.

By using this I will present new explicit examples of self-affine measures whose dimension can be computed. These examples rely on new results by Hochman and Solomyak, Bourgain, and Benoist and Quint, regarding the Furstenberg measure.

Dimension conservation for fractals under linear maps (following Furstenberg)

We all know the dimension theorem for vector spaces which asserts that dim(Im F) + dim(Ker F) = dim (Domain). In the paper Ergodic fractal measures and dimension conservation, H. Furstenberg proved an analogous result for the Hausdorff dimension of images of fractals under linear mappings. In his proof Furstenberg uses a dynamical approach and applies tools from ergodic theory. In the lecture I will present some basic background material and then a sketch of the proof for Furstenberg's theorem.

Here is an exercise sheet.

A Brief Introduction to Shifts of Finite Type

This talk will be a brief introduction to shifts of finite type in one and two dimensions. The talk should be accessible to everyone. A rudimentary knowledge of Perron Frobenius theory will help.

Something to think about: What is the number of 0,1 sequences of length 200 such that no two 1-s are adjacent?