Large intersection properties in Diophantine approximation and dynamical systems

TL;DR

This paper explores the large intersection properties of sets in Diophantine approximation and dynamical systems, providing new insights into approximation rates, circle homeomorphisms, and Hamiltonian perturbations.

Contribution

It introduces novel large intersection results for approximation sets and applies these to Diophantine, circle dynamics, and Hamiltonian system perturbations.

Findings

Established large intersection properties for approximation sets

Applied results to circle homeomorphisms

Extended analysis to Hamiltonian system perturbations

Abstract

We investigate the large intersection properties of the set of points that are approximated at a certain rate by a family of affine subspaces. We then apply our results to various sets arising in the metric theory of Diophantine approximation, in the study of the homeomorphisms of the circle and in the perturbation theory for Hamiltonian systems.