A Roth‐type theorem for dense subsets of Rd
Bibliographic record
Abstract
Let 1 < p < ∞ , p ≠ 2 . We prove that if d ⩾ d p is sufficiently large, and A ⊆ R d is a measurable set of positive upper density then there exists λ 0 = λ 0 ( A ) such that for all λ ⩾ λ 0 there are x , y ∈ R d such that { x , x + y , x + 2 y } ⊆ A and | | y | | p = λ , where | | y | | p = ( ∑ i | y i | p ) 1 / p is the l p ( R d ) -norm of a point y = ( y 1 , … , y d ) ∈ R d . This means that dense subsets of R d contain 3-term progressions of all sufficiently large gaps when the gap size is measured in the l p -metric. This statement is known to be false in the Euclidean l 2 -metric as well as in the l 1 and ℓ ∞ -metrics. One of the goals of this note is to understand this phenomenon. A distinctive feature of the proof is the use of multilinear singular integral operators, widely studied in classical time-frequency analysis, in the estimation of forms counting configurations.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.007 | 0.022 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.006 | 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".