A non-linear version of Bourgain’s projection theorem
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Bibliographic record
Abstract
We prove a version of Bourgain’s projection theorem for parametrized families of C^2 maps, which refines the original statement even in the linear case by requiring non-concentration only at a single natural scale. As one application, we show that if A is a Borel set of Hausdorff dimension close to 1 in \mathbb{R}^2 or close to 3/2 in \mathbb{R}^3 , then for y\in A outside of a very sparse set, the pinned distance set \{|x-y|:x\in A\} has Hausdorff dimension at least 1/2+c , where c is universal. Furthermore, the same holds if the distances are taken with respect to a C^2 norm of positive Gaussian curvature. As further applications, we obtain new bounds on the dimensions of spherical projections, and an improvement over the trivial estimate for incidences between \delta -balls and \delta -neighborhoods of curves in the plane, under fairly general assumptions. The proofs depend on a new multiscale decomposition of measures into “Frostman pieces” that may be of independent interest.
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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 | 0.005 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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