Birkhoff Measures, Birkhoff Sums, and Discrepancies

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.