Controlled Divergence of Discrepancy Sums

TL;DR

This paper demonstrates how fixing an irrational number's continued fraction expansion can control the growth rate of discrepancy sequences in rotation orbits, addressing an open question in discrepancy theory.

Contribution

It introduces a method to prescribe the growth rate of discrepancy sequences for all points in [0,1) by fixing the continued fraction expansion of the irrational rotation parameter.

Findings

Discrepancy sequences can be controlled to grow at a prescribed rate.

The method applies uniformly to all points in the interval.

It answers an open question posed by K. Park.

Abstract

Answering an informal question of K. Park, we show that by fixing some irrational alpha to have a particular standard continued fraction expansion, we may force the associated discrepancy sequences for all x in [0,1), which track the difference between the number of values in the orbit of x under rotation by alpha (modulo one) less than one half versus the number larger than one half, to have maximal values which grow at a prescribed rate.