Birkhoff Measures, Birkhoff Sums, and Discrepancies

TL;DR

This paper analyzes the distribution of points generated by irrational rotations on the circle, connecting Birkhoff sums, discrepancy, and the shape of associated measures, with new proofs and computational insights.

Contribution

It establishes a link between the support of Birkhoff measures and discrepancy, characterizes their shape for continued fraction denominators, and provides new proofs for classical results.

Findings

Support length of Birkhoff measures relates to discrepancy.

Graph of measures approximates an isosceles trapezoid for continued fraction denominators.

New proofs enable efficient computation of sums and discrepancies.

Abstract

We study the distribution of a sequence of points in the circle generated by rotations by a fixed irrational number $\rho$ with initial condition $x_0$, that is: $\{x_0+i\rho\}_{i=1}^n$. The \emph{discrepancy} as defined by Pisot and Van Der Corput \cite{VdCP}, quantifies how evenly distributed such a sequence is. Consider the ergodic or Birkhoff sum of mean zero $S(\rho,n,x):=\sum_{i=1}^{n} (\{x+i\rho\}-1/2)$, where $\{\cdot\}$ denotes the fractional part. This is a piecewise-linear map in the variable $x$ with $n$ branches, each with slope $n$. For fixed $n$ and $\rho$, let $\nu(\rho,n,z)$ be the number of pre-images of $S(\rho,n,x)=z$ divided by $n$. Then $\nu(\rho,n,z)$ is a probability density. We call the associated measures Birkhoff measures. We investigate how the graph of $\nu(\rho,n,z)$ varies with $n$. We prove that the length of the support of the Birkhoff measure $\nu(\rho,n,z)dz$ can be expressed in terms of the discrepancy. We also show that if $n$ is a continued fraction denominator of $\rho$, then the graph of $\nu(\rho,n,z)$ an approximate isosceles trapezoid. We also give new, brief, proofs of two classical results, one by Ramshaw \cite{Ramshaw} and one found by Kuipers-Niederreiter \cite{KN}. These results allow efficient computation of both Birkhoff sums and discrepancies.