Simultaneous rational approximation to successive powers of a real number
Bibliographic record
Abstract
We develop new tools leading, for each integer n ≥ 4 n\ge 4 , to a significantly improved upper bound for the uniform exponent of rational approximation λ ^ n ( ξ ) \widehat {\lambda }_n(\xi ) to successive powers 1 , ξ , … , ξ n 1,\xi ,\dots ,\xi ^n of a given real transcendental number ξ \xi . As an application, we obtain a refined lower bound for the exponent of approximation to ξ \xi by algebraic integers of degree at most n + 1 n+1 . The new lower bound is n / 2 + a n + 4 / 3 n/2+a\sqrt {n}+4/3 with a = ( 1 − log ( 2 ) ) / 2 ≃ 0.153 a=(1-\log (2))/2\simeq 0.153 , instead of the current n / 2 + O ( 1 ) n/2+\mathcal {O}(1)
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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 | 0.003 | 0.018 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.005 | 0.007 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.024 | 0.012 |
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".