Metric theory of continued fractions
Bibliographic record
Abstract
The examples we saw in Chapters 1 and 2 suggest that real numbers are arithmetically quite diverse. The theory of continued fractions as we have developed it allows us to recognise whether a given real number is rational or is a quadratic irrational; for the latter as well as for several transcendental numbers such as e , whose quotients follow a clear periodic pattern, we have precise knowledge of the quality of their rational approximations, as for instance in (2.40). A standard counting argument, however, shows that the totality of such numbers is countable; hence they form a subset of measure zero of the reals. It is therefore reasonable to look into the arithmetic properties of other real numbers – in particular, of almost all real numbers (of course, in the sense of the usual Lebesgue measure M). The classical problems of metric number theory include determining the measure of the set of numbers that satisfy a given arithmetic property. In the context of continued fractions, for example, we may ask about the measure of the set of numbers whose 100th quotient a 100 is exactly 100, or whose 100th convergent p n /q n satisfies q n < 10 10 . This is exactly the sort of question that we will address in this chapter.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
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".