Multi-rotations on the unit circle

TL;DR

This paper generalizes the classical density result of irrational rotations on the unit circle to multi-rotation orbits along finitely recurrent sequences, exploring their dimensions and Diophantine approximation connections.

Contribution

It extends the density theorem to multi-rotation orbits with finitely recurrent sequences and improves related dimension results for algebraic rotation parameters.

Findings

Multi-rotation orbits are dense under certain conditions.

Established connections between orbit dimensions and Diophantine approximation.

Improved previous results for algebraic rotation parameters.

Abstract

In this paper, we study multi-rotation orbits on the unit circle. We obtain a natural generalization of a classical result which says that orbits of irrational rotations on the unit circle are dense. It is possible to show that this result holds true if instead of iterating a single irrational rotation, one takes a multi-rotation orbit along a finitely recurrent sequence over finitely many different irrational rotations. We also discuss some connections between the box dimensions of multi-rotation orbits and Diophantine approximations. In particular, we improve a result by Feng and Xiong in the case when the rotation parameters are algebraic numbers.