Abstract
Non-degeneracy was first defined for hyperplanes by Elekes–Tóth, and later extended to spheres by Apfelbaum–Sharir: given a set P of m points in \(\mathbb {R}^d\) and some \(\beta \in (0,1)\), a \((d-1)\)-dimensional sphere (or a \((d-1)\)-sphere) S in \(\mathbb {R}^d\) is called \(\beta \)-nondegenerate with respect to P if S does not contain a proper subsphere \(S'\) such that \(|S'\cap P|\ge \beta |S\cap P|\). Apfelbaum–Sharir found an upper bound for the number of incidences between points and nondegenerate spheres in \(\mathbb {R}^3\), which was recently used by Zahl to obtain the best known bound for the unit distance problem in three dimensions. In this paper, we show that the number of incidences between m points and n \(\beta \)-nondegenerate 3-spheres in \(\mathbb {R}^4\) is \(O_{\beta ,\varepsilon }(m^{{15}/{19}+\varepsilon }n^{{16}/{19}}+mn^{{2}/{3}})\). As a consequence, we obtain a bound of \(O_{\varepsilon }(n^{2+4/11+\varepsilon })\) on the number of similar triangles formed by n points in \(\mathbb {R}^4\), an improvement over the previously best known bound \(O(n^{2+2/5})\). While proving this, we find it convenient to work with a more general definition of nondegeneracy: a bipartite graph \(G=(P,Q)\) is called \(\beta \)-nondegenerate if \(|N(q_1)\cap N(q_2)|<\beta |N(q_1)|\) for any two distinct vertices \(q_1,q_2\in Q\); here N(q) denotes the set of neighbors of q and \(\beta \) is some positive constant less than 1. A \(\beta \)-nondegenerate graph can have up to \(\Theta (|P||Q|)\) edges without any restriction, but must have much fewer edges if the graph is semi-algebraic or has bounded VC-dimension. We show that Elekes–Tóth’s bound for nondegenerate hyperplanes, Apfelbaum–Sharir’s bound for nondegenerate spheres in \(\mathbb {R}^3\), and our new bound for nondegenerate spheres in \(\mathbb {R}^4\), all hold under this new definition.
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Notes
What they actually proved is that the maximum number of \(\beta \)-nondegenerate, k-rich (i.e., containing at least k points of P) hyperplanes is \(O_{\beta ,d}({m^{d+1}}/{k^{d+2}}+{m^{d-1}}/{k^{d-1}})\) for \(\beta <\beta _d\) for some small \(\beta _d\). It is later shown in [23] that we can indeed let \(\beta _d=1\). See [7] for how this implies (1).
via the map \((x_1,\dots , x_d)\mapsto (x_1,\dots , x_d, x_1^2+\dots +x_d^2)\)
To see this bound is indeed an improvement from \((mn)^{4/5}+mn^{2/3}\): clearly \(mn^{1/2}\le mn^{2/3}\); \(m^{8/11}n^{9/11}\le (mn)^{4/5}\) iff \(n\le m^4\), which holds true by Claim 1.8.
defined at the beginning of the proof of Theorem 1.4
As observed in [14], it does not matter whether Q belongs to \(\mathbb {R}^s\) or a s-dimensional variety of bounded degree in a larger space
The image of a sphere in this projection is either an ellipsoid or a solid ellipse. This is similar to if we project a circle in \(\mathbb {R}^3\) to plane we either get an ellipse or a line segment.
There is a stronger point-circle incidence bound in [24], but this one has fewer terms and is enough for our purpose.
If \(\Delta \) has more than one longest edges, pick either one and the calculation may be off by at most a constant factor.
The other case can be treated similarly by rewriting \(I_3\) as incidences between \(|S_{a,dr}|\) 2-spheres and \(|S_{a,r}|\) points on \(S_{a,r}\).
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Acknowledgements
The author would like to thank Larry Guth and Micha Sharir for helpful conversations. She also thanks Ethan Jaffe, Vishesh Jain, and Jake Wellens for some helpful comments. Finally, the author is very grateful for the referee for many helpful comments.
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Do, T. Nondegenerate Spheres in Four Dimensions. Discrete Comput Geom 68, 406–424 (2022). https://doi.org/10.1007/s00454-021-00366-5
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DOI: https://doi.org/10.1007/s00454-021-00366-5