Complex Continued Fractions:Theoretical Aspects of Hurwitz's Algorithm
Bibliographic record
Abstract
Inspired by their remarkable properties, mathematicians have sought in several contexts objects similar to regular continued fractions. Hurwitz Continued fractions (HCF), proposed by Adolf Hurwitz in 1887, give a natural analogue in the complex plane. The HCF of a complex number ζ is a sequence of Gaussian integers obtained from a specific algorithm. Among the similarities between HCF and regular continued fractions we can find the irrationality of a complex number as a necessary and sufficient condition for its HCF to be infinite. In this case, there even are combinatorial conditions on the HCF implying transcendence. The usual characterization of quadratic surds in terms of periodic continued fractions also holds. However, É. Galois' theorem on purely periodic continued fractions is not true unless additional restrictions are imposed. HCF can be understood through a dynamical perspective. With a suitably defined Gauss map, HCF give rise to an ergodic dynamical system. Although the corresponding shift space has a complicated structure, some classical Hausdorff dimension results still hold. For example, the set of complex numbers with a bounded HCF has full Hausdorff dimension (proved by V. Jarník in 1929 for the real line) and the Hausdoff dimension of the set of complex numbers with a HCF tending to ∞ is half of that of the ambient space's (proved by I.J. Good in 1941 for the real line). The theory of HCF is far from being completed, and its similarities and discrepancies with their real counterpart stem new problems. The present thesis is a survey of new and old results on HCF. In the preface, a detailed list of all the novel aspects is given. Additionally, three original research papers are included as appendices. The first two deal with HCF theory. The third one --- applying the geometry of regular continued fractions--- contains a simple proof of P. Bengoechea's quantitative improvement of Serret's Theorem on equivalent numbers.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".