Computational study of irrational rotations via exact discontinuity tracking

TL;DR

This work addresses the problem of computing the discrepancy $D_N(x,\rho)$ for irrational rotations and its associated pdf with machine-precision accuracy. It introduces a discontinuity-tracking algorithm that exactly defines $D_N$ by following its jump discontinuities, reducing the computation to $O(N)$ time and $O(N)$ storage, and enables $O(N\log N)$ pdf computation. The approach yields precise statistics such as the support $\|D_N\|_{\infty}$, the variance $||D_N||_2^2$, and the kurtosis, and reveals self-similar patterns tied to the Diophantine properties of $\rho$, including spikes in the pdf when $N$ is a multiple of convergent denominators. The results provide a powerful tool for experimental mathematics in this area, allowing rapid generation of exact figures and testing of conjectures across a wide range of irrationals. The work also outlines future extensions to other discrepancy sums and connections to the three-gap theorem, promising broader impact in numerical analysis and dynamical systems.

Abstract

The discrepancy sum $D_N(x,ρ)$ for irrational rotations has been of interest to mathematicians for over a century. While historically studied in an ``almost-everywhere'' or asymptotic sense, $D_N$ for finite N is increasingly an object of interest for its nontrivial properties that depend on the Diophantine properties of $ρ$. This behavior is periodic in N with respect to the quotients of the continued fraction convergents, which grow quickly for some irrationals. Thus the stable computation of the sum is necessary for forming conjectures about its properties. However, computing the exact value of the sum and its corresponding probability density function (pdf) is notoriously difficult due to numerical instability in the sum itself and the failure of sampling methods to capture its jump discontinuities. This paper presents a novel computational algorithm that fully defines the discrepancy function and its associated pdf through its discontinuities. This allows the calculation of $D_N(x,ρ)$ to machine precision with minimal storage in O(N) time. This vast improvement in computability over the O(N^2) naive version enables, for the first time, the direct computation of the exact pdf up to machine precision in O(N log N) time, and with it, key properties of the discrepancy: $ \|D_N \|_{\infty}$ (half of the support of the pdf), $ \|D_N \|_{2}^2$ (the variance of the pdf), and the kurtosis of the pdf. A key strength of the algorithm lies in its ability to produce clear, exact figures, allowing the development of mathematical intuition and the quick testing of conjectures. As an example, a newly conjectured pattern is presented: when $ρ$ is well-approximated by rational $\frac{p_n}{q_n}$, the pdf exhibits a predictable spiked-trapezoidal pattern when $N=kq_n$. These shapes degrade as $k$ increases, at a speed depending on how well $\frac{p_n}{q_n}$ approximates $ρ$.

Figures (17)

• Figure 1: This figure shows the plot of $D_N(x,L)$ for different $N$. The three rows show different levels of a self-similar structure changing with $N$.
• Figure 2: (Left) $\text{pdf}$ for $\rho= \frac{\sqrt{5}-1}{2}$ (the golden ratio) for two choices of $N$. The top, $N=329$, shows a few straight platforms connected by very short segments of varying heights, while $N=610$ (a partial quotient for $\rho$) has a long straight horizontal line at $y=1$, indicating that all branches pass through a particular range of values containing zero, with piecewise-constant line segments making up the sides of this trapezoid-like distribution. (Right) The $\text{pdf}$ functions are zoomed in to show the detail the algorithm is able to capture.
• Figure 3: (Left) $\text{pdf}$ for $\rho =L$ (Liouville's constant). Two consecutive values of $N$ are shown, $N=300$, a multiple of the second convergent denominator, and $N=301$, where the support of the distribution immediately spreads out. (Right) The $\text{pdf}$ functions are zoomed in to show the detail the algorithm is able to capture.
• Figure 4: Comparison of time complexities for the algorithms.
• Figure 5: The maximum error ($\infty$-norm) of the algorithm grows with the number of terms as $\frac{1}{2} \varepsilon_{\text{mach}}$.