An old conjecture of Erdos–Turán on additive bases
Bibliographic record
Abstract
There is a 1941 conjecture of Erdős and Turán on what is now called additive basis that we restate: Conjecture 0.1 (Erdős and Turán) . Suppose that 0 = δ 0 > δ 1 > δ 2 > δ 3 ⋯ 0 = \delta _0>\delta _1>\delta _2>\delta _3\cdots is an increasing sequence of integers and \[ s ( z ) := ∑ i = 0 ∞ z δ i . s(z) : = \sum _{i=0}^\infty z^{\delta _i}. \] Suppose that \[ s 2 ( z ) := ∑ i = 0 ∞ b i z i . s^2(z) := \sum _{i=0}^\infty b_i z^i. \] If b i > 0 b_i>0 for all i i , then { b n } \{b_n\} is unbounded. Our main purpose is to show that the sequence { b n } \{b_n\}
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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 | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.013 | 0.003 |
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".