From a Packing Problem to Quantitative Recurrence in $[0,1]$ and the Lagrange Spectrum of Interval Exchanges

TL;DR

This paper establishes optimal recurrence constants for Lebesgue measure-preserving interval maps and the Lagrange spectrum of interval exchanges, using a novel plane packing problem with a pseudo-norm.

Contribution

It introduces a new approach linking packing problems in the plane to recurrence properties and Lagrange spectra in interval exchange transformations.

Findings

Derived optimal recurrence constants for interval maps.

Characterized the Lagrange spectrum for interval exchanges.

Connected packing problems to dynamical recurrence and Diophantine approximation.

Abstract

This article provides optimal constants for two quantitative recurrence problems. First of all for recurrence of maps of the interval [0,1] that preserve the Lebesgue measure and on the other hand Lagrange spectrum of interval exchange transformations. Both results are based on a non-conventional packing problem in the plane with respect to the "pseudo-norm" N(x,y) = sqrt(|xy|).