Filling times for linear flow on the torus with truncated Diophantine conditions: a brief review and new proof

TL;DR

This paper extends the geometry-of-numbers method to analyze filling times for linear flows on the torus with truncated Diophantine conditions, recovering optimal estimates and reviewing related dynamics and applications.

Contribution

It introduces an extension of the geometry-of-numbers approach to truncated Diophantine conditions, providing a new proof of optimal filling time estimates.

Findings

Extended the geometry-of-numbers method to truncated Diophantine conditions.

Recovered optimal filling time estimates originally obtained in 2003.

Reviewed the dynamics and applications of linear flows on the torus.

Abstract

We show that the geometry-of-numbers method used by A. Bounemoura to obtain filling times for linear flow on the torus satisfying Diophantine conditions may be extended to the case of linear flow with truncated Diophantine conditions, and we use these methods to recover the optimal estimate first obtained by M. Berti, L. Biasco, and P. Bolle in 2003. We also briefly review the dynamics of linear flow on the torus, previous results, optimality, and applications of these estimates.