Published In
Experimental Mathematics
Document Type
Article
Publication Date
10-2026
Subjects
Irrational rotation, discrepancy, Birkhoff sum, diophantine approximation, uniform distribution modulo one
Abstract
We present an algorithm for studying the sum๐บ๐โก(๐ฅ,๐)=๐โ๐=1[{๐ฅ+๐โข๐}โ12],
which is closely related to the discrepancy of the irrational rotation. Computation of๐บ๐and associated statistics is notoriously difficult due to numerical instability and the failure of sampling to capture the discontinuities. Our algorithm instead fully defines๐บ๐through its discontinuities, which allows computation of๐บ๐โก(๐ฅ,๐)with errors on the order of๐times machine epsilon in๐โก(๐)time for practical๐. The associated probability density function can then be computed in๐โก(๐โขlogโก๐)time. Presented as a tool for experimental mathematics, the algorithm enables reliable plots of the descriptive statistics of๐บ๐over๐, in particular the length of the support of the pdf (the discrepancy), revealing extremal behavior related to the arithmetic properties of๐. We also observe a new pattern: when๐is well approximated by a continued-fraction convergent๐๐๐๐, the pdf exhibits a predictable spiked-trapezoidal pattern when๐=๐โข๐๐, eventually degrading at a rate that appears to depend on approximation quality.
Rights
Copyright (c) 2026 The Authors
This work is licensed under a Creative Commons Attribution 4.0 International License.
Locate the Document
DOI
10.1080/10586458.2026.2742231
Persistent Identifier
https://archives.pdx.edu/ds/psu/45226
Citation Details
Kravitz, H. (2026). Computational Study of Irrational Rotations via Exact Discontinuity Tracking: A New Experimental Framework. Experimental Mathematics, 1-18.