Incidences with $k$-non-degenerate sets and their applications
Why this work is in the frame
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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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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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