Birkhoff Measures, Birkhoff Sums, and Discrepancies

TL;DR

The paper investigates how the orbit of an irrational circle rotation, encoded through Birkhoff sums $S(\rho,n,x)$, distributes on the circle by introducing Birkhoff measures $\nu(\rho,n,z)$. It establishes a precise link between the support of these measures and discrepancy via $\text{supp}(\nu)=nD_n$, and shows a precise trapezoidal structure at continued fraction convergents. The authors provide new, concise proofs of classical Ramshaw-type results and Kuipers–Niederreiter findings, along with recursion formulas based on Ostrowski expansions that reveal a rich fractal-parabolic geometry of Birkhoff sums. The work also includes numerical demonstrations and discusses implications for efficient computation of both Birkhoff sums and discrepancies, highlighting the intricate interplay between rotation dynamics, discrepancy theory, and measure-theoretic representations.

Abstract

We study the distribution of a sequence of points in the circle generated by rotations by a fixed irrational number $ρ$ with initial condition $x_0$, that is: $\{x_0+iρ\}_{i=1}^n$. 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 $S(ρ,n,x):=\sum_{i=1}^{n} (\{x+iρ\}-1/2)$, 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 $ρ$, let $ν(ρ,n,z)$ be the number of pre-images of $S(ρ,n,x)=z$ divided by $n$. Then $ν(ρ,n,z)$ is a probability density. We call the associated measures Birkhoff measures. We investigate how the graph of $ν(ρ,n,z)$ varies with $n$. We prove that the length of the support of the Birkhoff measure $ν(ρ,n,z)dz$ can be expressed in terms of the discrepancy. We also show that if $n$ is a continued fraction denominator of $ρ$, then the graph of $ν(ρ,n,z)$ 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.

Figures (8)

• Figure 1.1: The construction of the Birkhoff measure for $\rho$ equal to the golden mean with $n=13$.
• Figure 1.2: For $\rho$ the golden mean, we show the range of $S(\rho,n,x)$, indicated by the min/max values for the sum $S$ over $x\in [0,1)$ with increasing value of $n$. The plot also shows $S(\rho,n,x_0)$ for two initial conditions indicated, $x_0=0$ and $x_0=1/\sqrt{5}$.
• Figure 2.1: The sum of seven translates of the Birkhoff measure associated with $e-2$ and $n=2024$.
• Figure 4.1: Sketch of the branches of $S$ with $d=0$ in red and with $d>0$ in blue. Here, $S_j$ is the third branch. The lengths of the segments indicated by $A$, $B$, and $C$, are, respectively, $\frac{(q+1)d}{2}$, $\frac{i_+d}{q}$, and $\frac{(q+1-2i_+)d}{2}$.
• Figure 4.2: Illustration of how changing the reduced residue in $\rho$ simplifies the construction of $\nu(\rho,q,z)$. We first construct $\nu(e-2,32,z)$. knowing that $23/32$ is an odd approximant of $e-2$, we then construct $\nu(e-2-22/32, 32,z)$. Note that there is 1 longer interval and 31 shorter ones.