Computational study of irrational rotations via exact discontinuity tracking

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 $ρ$.