Bibliographic record
Abstract
Arithmetic combinatorics, or additive combinatorics, is a fast developing area of research combining elements of number theory, combinatorics, harmonic analysis and ergodic theory.Its arguably best-known result, and the one that brought it to global prominence, is the proof by Ben Green and Terence Tao of the long-standing conjecture that primes contain arbitrarily long arithmetic progressions.There are many accounts and expositions of the Green-Tao theorem, including the articles by Kra [119] and Tao [182] in the Bulletin.The purpose of the present article is to survey a broader, highly interconnected network of questions and results, built over the decades and spanning several areas of mathematics, of which the Green-Tao theorem is a famous descendant.An old geometric problem lies at the heart of key conjectures in harmonic analysis.A major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles.An unexpected argument points harmonic analysts towards additive number theory, with consequences that could have hardly been anticipated.We will not try to give a comprehensive survey of harmonic analysis, combinatorics, or additive number theory.We will not even be able to do full justice to our specific areas of focus, instead referring the reader to the more complete expositions and surveys listed in Section 7. Our goal here is to emphasize the connections between these areas; we will thus concentrate on relatively few problems, chosen as much for their importance to their fields as for their links to each other.The article is written from the point of view of an analyst who, in the course of her work, was gradually introduced to the questions discussed here and found them fascinating.We hope that the reader will enjoy a taste of this experience.1.The Kakeya problem 1.1.Life during wartime.By all accounts, Abram Samoilovitch Besicovitch (1891-1970) had an interesting life.He was born in Berdyansk, in the south of
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.007 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.011 | 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".