Moebius Disjointness

The Mobius function is considered a highly random function, whose randomness is reflected through many theorems in number theory such as PNT and PNT in APs.

Recently, P. Sarnak has conjectured a generalization of those theorems, interpreted suitably via dynamics, claiming that the Mobius flow is disjoint from any entropy-0 flow. We will give an introduction to the problem and discuss a recent result due to Bourgain-Sarnak-Ziegler showing disjointness of the Mobius flow from the horocyclic flow on the modular surface.

Nice introductory article by Sarnak is given here and an introductory computer presentation is here.

Limiting distributions of translates of divergent diagonal orbits

We define a natural topology on the collection of (equivalence classes up to scaling of) locally finite measures on a homogeneous space and prove that under this topology, pushforwards of certain infinite volume orbits equidistribute in the ambient space. As an application of our results we prove an asymptotic formula for the number of integral points on some varieties as the radius goes to infinity. Joint work with Uri Shapira.

Non-divergence in SO(n,1) and diophantine approximation on spheres.

The goal of the talk will be to explain quantitative non-divergence estimates in the spirit of Kleinbock-Margulis, but in the easier case of the rank-one groups SO(n,1). This will be used to study approximation of points on the sphere S^{n-1} of dimension n-1 by rational points on S^{n-1}.

Equidistribution of finite continued fractions

In Kaplum 319 NOTE UNUSUAL LOCATION

Continued fraction expansion (CFE) is a presentation of numbers which is closely related to Diophantine approximation and other number theoretic fields. It is well known that for almost every x in (0,1), the coefficients appearing in the CFE of x obey the Gauss-Kuzmin statistics. This claim is not true for all x, and in particular it is not true for rational numbers which have finite CFE. In order to still have some statistical law, we instead group together the rationals p/q in (0,1) for q fixed and (p,q)=1 and ask whether their combined statistics converges as q goes to infinity. In this talk I will define what are continued fractions, why they are useful and how to understand the result above as an equidistribution of an orbit of the Gauss map. As we shall see, it is quite interesting to know when this equidistribution result fails and how to "fix" it. In particular I will show how to formulate and solve the problem of equidistribution of finite continued fractions using the language of dynamics of lattices in SL_2(Z)\SL_2(R). This will in turn imply a stronger statistical law of the CFE of rational numbers.

This is a joint work with Uri Shapira from the Technion.

Obvious divergent trajectories in Q-algebraic groups

In the theory of Diophantine approximations, singular points are ones for which Dirichlet's theorem can be infinitely improved. It is easy to see that all rational points are singular. In the special case of dimension one, the only singular points are the rational ones. In higher dimensions, points lying on a rational hyperplane are also obviously singular. However, in this case there are additional singular points. In the dynamical setting the singular points are related to divergent trajectories. In the talk I will define obvious divergent trajectories and explain the relation to rational points. In addition, I will present the more general setting involving Q-algebraic groups. Lastly I will discuss results concerning classification of divergent trajectories in Q-algebraic groups.

Certain cut and project sets, their subsets and projections

Distribution of sublattices and of projections of a lattice has been studied by W Schmidt in [1]. Projections of quasicrystals are often non-discrete (in any directions), and usually quasicrystals don't have nice "sub-quasicrystal" structures. However, Penrose tiling [2] and other similar quasicrystals have a sub-quasilattice structure, in fact one can easily build a quasiperiodic point set on a plane in which every non-trivial intersection with a line is quasiperiodic. We'll define cut and project sets, look at a family of cut and project sets which have this nice structure, prove that the sub-cutandproject sets (and projections) form a nice family of point sets. J. Markloff has shown that on such a family there exists a natural measure, and parametrisation. We'll calculate the distribution of shapes of the sub-cutandproject sets with respect to that parametrisation.

For more information on the theorem for lattices, please see the original paper of Schmidt and for more about the Penrose case, see the paper of De Bruijn.

Equidistribution of Iwasawa components of lattices and asymptotic properties of primitive vectors

I will discuss questions of counting lattice points in domains defined w.r.t. Iwasawa coordinates in rank-one groups and in SL(3,R). These counting results imply equidistribution of Iwasawa components of lattices. When considering integral lattices, i.e. SL(2,Z) or SL(3,Z), the Iwasawa components represent parameters related to primitive integral vectors, such as their directions, the integral lattices in their orthogonal space, or the shortest solution to their associated gcd equation. The equidistribution of the Iwasawa components of integral lattices thus implies the equidistribution of these parameters.