On the image in the torus of sparse points on expanding analytic curves

Abstract: It is known that the projection to the 2-torus of the normalised parameter measure on a circle of radius $R$ in the plane becomes uniformly distributed as $R$ grows to infinity. I will discuss the following natural discrete analogue for this problem. Starting from an angle and a sequence of radii {$R_n$} which diverges to infinity, I will consider the projection to the 2-torus of the n’th roots of unity rotated by this angle and dilated by a factor of $R_n$. The interesting regime in this problem is when $R_n$ is much larger than n so that the dilated roots of unity appear sparsely on the dilated circle.I will discuss 3 types of results:

Validity of equidistribution for all angles when the sparsity is polynomial. Failure of equidistribution for some super polynomial dilations. Equidistribution for almost all angles for arbitrary dilations. I will discuss the above type of results in greater generality and I will try to explain how the theory of o-minimal structures is related to the proof.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar