Speaker: Michael Bersudsky
18 November, 2020 14:30 UTC
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 then pass to discuss more general results on the projection to the $d$-torus of dilations of varying analytic curves in $d$-space.