On the binary expansions of algebraic numbers
Bibliographic record
Abstract
Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1’s in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D > 1 , then the number # ( | y | , N ) of 1-bits in the expansion of | y | through bit position N satisfies # ( | y | , N ) > C N 1 / D for a positive number C (depending on y ) and sufficiently large N . This in itself establishes the transcendency of a class of reals ∑ n ≥ 0 1 / 2 f ( n ) where the integer-valued function f grows sufficiently fast; say, faster than any fixed power of n . By these methods we re-establish the transcendency of the Kempner–Mahler number ∑ n ≥ 0 1 / 2 2 n , yet we can also handle numbers with a substantially denser occurrence of 1’s. Though the number z = ∑ n ≥ 0 1 / 2 n 2 has too high a 1’s density for application of our central result, we are able to invoke some rather intricate number-theoretical analysis and extended computations to reveal aspects of the binary structure of z 2 .
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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 | 0.029 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.009 | 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".