I'll report on recent applications of the LLL to two problems in the dynamics of general countable groups. The first concerns the existence of free symbolic minimal actions; the second asks about realizations of URS's (uniformly recurrent subgroups) as stability systems. I'll then concentrate on a solution of Gabor Elek to the second problem in the case of finitely generated groups.

I'll report on recent applications of the LLL to two problems in the dynamics of general countable groups. The first concerns the existence of free symbolic minimal actions; the second asks about realizations of URS's (uniformly recurrent subgroups) as stability systems. I'll then concentrate on a solution of Gabor Elek to the second problem in the case of finitely generated groups.

A dimension gap for continued fractions with independent digits (after Kifer, Peres and Weiss)

I'll report on recent applications of the LLL to two problems in the dynamics of general countable groups. The first concerns the existence of free symbolic minimal actions; the second asks about realizations of URS's (uniformly recurrent subgroups) as stability systems. I'll then concentrate on a solution of Gabor Elek to the second problem in the case of finitely generated groups.

I'll report on recent applications of the LLL to two problems in the dynamics of general countable groups. The first concerns the existence of free symbolic minimal actions; the second asks about realizations of URS's (uniformly recurrent subgroups) as stability systems. I'll then concentrate on a solution of Gabor Elek to the second problem in the case of finitely generated groups.

I'll report on recent applications of the LLL to two problems in the dynamics of general countable groups. The first concerns the existence of free symbolic minimal actions; the second asks about realizations of URS's (uniformly recurrent subgroups) as stability systems. I'll then concentrate on a solution of Gabor Elek to the second problem in the case of finitely generated groups.

I'll report on recent applications of the LLL to two problems in the dynamics of general countable groups. The first concerns the existence of free symbolic minimal actions; the second asks about realizations of URS's (uniformly recurrent subgroups) as stability systems. I'll then concentrate on a solution of Gabor Elek to the second problem in the case of finitely generated groups.

Abstract:

It is an open question whether the fractional parts of nonlinear polynomials at integers have the same fine-scale statistics as a Poisson point process. We provide explicit Diophantine conditions on the coefficients of degree 2 polynomials under which the limit of an averaged pair correlation density is consistent with the Poisson distribution, using a recent effective Ratner equidistribution result on the space of affine lattices due to Strömbergsson. This is joint work with Jens Marklof.

It is well known that for almost every x in (0,1) its orbit under the Gauss map, namely T(x)=1/x-[1/x], equidistributes with respect to the Gauss-Kuzmin measure. This claim is not true for all x, and in particular it is not true for rational numbers which have finite "orbits" which terminate in 0. In order to still have some equidistribution, we instead group together the orbits corresponding to p/q when q is fixed and (p,q)=1 and ask whether these finite sets equidistribute as q goes to infinity.