On the Frequency of Balanced Times in Cylinder Flows

TL;DR

This paper studies the distribution of balanced times in irrational rotations, showing that for most pairs, the sum over reciprocals diverges, but some special cases exhibit convergence, revealing nuanced distribution properties.

Contribution

It introduces a detailed analysis of the sparseness of balanced times in cylinder flows, highlighting divergence and convergence behaviors for generic and exceptional parameters.

Findings

For generic (alpha,x), the sum over reciprocals diverges.

Certain exceptional alpha values lead to convergent sums for all x.

The results reveal intricate distribution patterns of balanced times.

Abstract

Given an irrational alpha and an x in the unit interval, the set of balanced times, for which the same number of (k*alpha+x) (modulo one) are less than or equal to one half as are larger than one half, is in general infinite, but sparse in terms of density. We investigate the sparseness of this sequence in terms of summation over reciprocals. Our results are that for the generic pair (alpha,x), the resulting sum diverges, but there are certain exceptional alpha for which the associated sums converge for every x.