Albert Artiles | Searcharxiv

arXiv2025

The Return Map of the Cross Section of Horizontally Short Lattice Surfaces is Weakly Mixing

We prove that the return map of the unstable horocycle flow on the space of horizontally short translation surfaces associated to a lattice surface $(X, ω)$ is weakly mixing. This extends a result of Cheung-Quas for the square torus to all lattice surfaces. The proof adapts their criterion for weakly mixing and uses quantitative bounds for Siegel-Veech transforms restricted to the Poincaré section of horizontally short surfaces.

math.DS↗

arXiv2023

The Minimal Denominator Function and Geometric Generalizations

We provide a geometric interpretation for a normalized version of the minimal denominator function, $$q_{\min}(x,δ)=\min\left\{q\in \mathbb{N}: \text{ there exists } p\in\mathbb{Z} \text{ such that } \frac{p}{q}\in (x-δ,x+δ)\right\},$$ introduced by Chen and Haynes. We use this interpretation to compute the limiting distribution of a suitably normalized version of $q_{\min}(x,δ)$ as a function of $x$, and give generalizations of the idea of minimal denominators to higher-dimensional unimodular lattices, linear forms, and translation surfaces. The key idea is to turn this circle of problems into equidistribution problems for translates of unipotent orbits of a Lie group action on an appropriate moduli space.

math.DS↗