Approximation of points in the plane by generic lattice orbits

TL;DR

This paper establishes bounds on how well points in the plane can be approximated by lattice orbits, providing uniform estimates for almost all such orbits, advancing understanding in Diophantine approximation.

Contribution

It introduces new bounds for Diophantine exponents related to lattice orbits in the plane, applicable uniformly to almost all orbits.

Findings

Derived upper bounds for approximation exponents.

Derived lower bounds for approximation exponents.

Bounds are uniform for almost all lattice orbits.

Abstract

We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $\Gamma<\mathrm{SL}_2(\mathbb{R})$ acting linearly on $\mathbb{R}^2$. Our method gives bounds that are uniform for almost all orbits.