Incidences with $k$-non-degenerate sets and their applications
Bibliographic record
Abstract
We study point-sphere and point-plane incidences in the three-dimensional space. In particular, for $1 < k < n$, we say that a set of spheres (resp., of planes) is $k$-non-degenerate if no circle is contained in $k$ spheres of the set (resp., if no line is contained in $k$ planes of the set). We prove that, for every $\varepsilon>0$, the number of incidences between a set of $m$ points and a $k$-non-degenerate set of $n$ spheres is \[ O(m^{3/4+\varepsilon}n^{3/4}k^{1/4}+n+mk).\] Similarly, we prove that, for every $\varepsilon>0$, the number of incidences between a set of $m$ points and a $k$-non-degenerate set of $n$ planes is \[ O(m^{4/5+\varepsilon}n^{3/5}k^{2/5} + n + mk). \] These bounds are obtained by using the polynomial partitioning technique, recently introduced by Guth and Katz. More specifically, in our proofs we use a pair of constant-degree partitioning polynomials. We also present a couple of applications of $k$-non-degenerate sets: (i) We consider an extension of the three-dimensional unit distances problem, in which we are given a set $D$ of $k$ distinct distances and ask for a three-dimensional set of $m$ points that maximizes the number of pairs of points that span a distance from $D$. By relying on $k$-non-degenerate sets of spheres, we prove an upper bound of $O(m^{236/149+\varepsilon}k^{125/149})$ for the problem (which improves the trivial bound for large values of $k$). Â Â Â (ii) We consider the maximum number of incidences between a three-dimensional set of $n$ planes (without any restrictions) and a set of $m$ points, such that no $k$ points are collinear. Our bound for $k$-non-degenerate planes immediately implies a bound of $O(n^{4/5+\varepsilon}m^{3/5}k^{2/5} + m + nk)$ for this problem, generalizing the previous bound $O(n^{4/5}m^{3/5} + n\log m)$ for the specific case where no three points are collinear (up to the $\varepsilon$ in the exponent).
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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 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.004 | 0.009 |
| Research integrity | 0.001 | 0.007 |
| Insufficient payload (model declined to judge) | 0.008 | 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".