A Roth‐type theorem for dense subsets of Rd
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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.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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 it