arXiv subjects

Elijah Fromm

Publications and source records attributed to Elijah Fromm.

4 recordsLinked to original sources

arXiv2023

Joint equidistribution of maximal flat cylinders and holonomies for Anosov homogeneous spaces

Let $G$ be a connected semisimple real algebraic group and $P<G$ be a minimal parabolic subgroup with Langlands decomposition $P=MAN$. Let $\Gamma < G$ be a Zariski dense Anosov subgroup with respect to $P$. Since $\Gamma$ is Anosov, the set of conjugacy classes of primitive elements of $\Gamma$ is in one-to-one correspondence with the set of (positively oriented) maximal flat cylinders in $\Gamma\backslash G/M$. We describe the joint equidistribution of maximal flat cylinders and their holonomies as their circumferences tend to infinity. This result can be viewed as the Anosov analogue of the joint equidistribution result of closed geodesics and holonomies in rank one by Margulis--Mohammadi--Oh.

math.DS↗

arXiv2022

Geodesic nets on flat spheres

We consider geodesic nets (critical points of a length functional on the space of embedded graphs) on doubled polygons (topological 2-spheres endowed with a flat metric away from finitely many cone singularities). We use the theorem of Gauss-Bonnet to demonstrate the existence and non-existence of specific geodesic nets on regular doubled polygons.

math.DG↗

arXiv2018

Tiling Billards on Triangle Tilings, and Interval Exchange Transformations

We consider the dynamics of light rays in triangle tilings where triangles are transparent and adjacent triangles have equal but opposite indices of refraction. We find that the behavior of a trajectory on a triangle tiling is described by an orientation-reversing three-interval exchange transformation on the circle, and that the behavior of all the trajectories on a given triangle tiling is described by a polygon exchange transformation. We show that, for a particular choice of triangle tiling, certain trajectories approach the Rauzy fractal, under rescaling.

math.DS↗

arXiv2017

Improving the Burgess bound via Polya-Vinogradov

We show that even mild improvements of the Polya-Vinogradov inequality would imply significant improvements of Burgess' bound on character sums. Our main ingredients are a lower bound on certain types of character sums (coming from works of the second author joint with J. Bober and Y. Lamzouri) and a quantitative relationship between the mean and the logarithmic mean of a completely multiplicative function.

math.NT↗