Ph.D Thesis | |

Ph.D Student | Bersudsky Michael |
---|---|

Subject | Equidistribution of Some Expanding Sets in Homogeneous Spaces |

Department | Department of Mathematics |

Supervisor | ASSOCIATE PROF. Uri Shapira |

Full Thesis text |

Motivated by many applications in number theory, a main line of research in homogeneous dynamics studies expanding subsets of a homogeneous space with the goal of describing their asymptotic behavior. In this thesis, we present two novel results in that direction.

In the first part of the thesis we prove the
equidistribution of a sequence of expanding periodic orbits in an S-arithmetic
space, which is used to compute the statistics of SL_{d}(**Z**)
matrices lying on level sets of an integral polynomial defined on SL_{d}(**R**);
a result that is a variant of the well-known theorem proved by Linnik about the
equidistribution of radially projected integral vectors from a large sphere
into the unit sphere. The main motivation behind this project is a
generalization of the work of Aka, Einsiedler and Shapira in various
directions. For example, we compute the joint distribution of the residue
classes modulo q and the properly normalized orthogonal lattices of primitive
integral vectors lying on the level set -x_{1}^{2}-x_{2}^{2}-x_{3}^{2}_{4}^{2}=N
as N grows to infinity, where the normalized orthogonal lattices sit in a
submanifold of the moduli space of rank-3 discrete subgroups of **R**^{4}.

In the second part of the thesis, we study the statistics
modulo one of sparse points on expanding analytic curves. It is known that the
image in **R**^{2}/**Z**^{2} of a circle of radius *r*
in the plane becomes equidistributed as *r *grows to infinity. We consider
the following sparse version of this phenomenon. Starting from a sequence of
radii {*r*_{n}}_{n>1} which diverges to infinity and an
angle *w*, we consider the projection to **R**^{2}/**Z**^{2
}of the n^{th} roots of unity rotated by angle *w *and
dilated by a factor of *r*_{n}. We prove that if *r*_{n}
grows polynomially in n, then the image of these sparse collections becomes
equidistributed, and moreover, if *r*_{n} grows arbitrarily fast,
then we show that equidistribution holds for almost all *w*.
Interestingly, we found that for any angle *w* there is a sequence of
radii growing to infinity faster than any polynomial for which equidistribution
fails dramatically. In greater generality, we prove this type of results for
dilations of varying analytic curves in **R**^{d}. A novel component
of the proof is the use of the theory of o-minimal structures to control
exponential sums.