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Incidences between points and non-coplanar circles | | Micha Sharir
; Adam Sheffer
; Joshua Zahl
; | | Date: |
1 Aug 2012 | | Abstract: | We establish an improved upper bound for the number of incidences between $m$
points and $n$ arbitrary circles in three dimensions. The previous best known
bound, which applies in any dimension, is $O^*(m^{2/3}n^{2/3} +
m^{6/11}n^{9/11}+m+n)$. Since all the points and circles may lie on a common
plane (or sphere), it is impossible to improve the three-dimensional bound
without improving the two-dimensional one.
Nevertheless, we show that if the set of circles is required to be "truly
three-dimensional" in the sense that there exists a $q<n$ so that no sphere or
plane contains more than $q$ of the circles, then the bound can be improved to
[O^*ig(m^{3/7}n^{6/7} + m^{2/3}n^{1/2}q^{1/6} + m^{6/11}n^{15/22}q^{3/22} +
m + nig).]
For various ranges of parameters (e.g., when $m=Theta(n)$ and $q =
o(n^{7/9})$), this bound is smaller than the best known two-dimensional lower
bound $Omega^*(m^{2/3}n^{2/3}+m+n)$. Thus we obtain an incidence theorem
analogous to the one in the recent distinct distances paper by Guth and Katz,
which states that if we have a collection of points and lines in $R^3$ and we
restrict the number of lines that can lie on a common plane or regulus, then
the maximum number of point-line incidences is smaller than the maximum number
of incidences that can occur in the plane.
Our result is obtained by applying the polynomial partitioning technique of
Guth and Katz using a constant-degree partitioning polynomial, as was also
recently used by Solymosi and Tao. We also rely on various additional tools
from analytic, algebraic, and combinatorial geometry. | | Source: | arXiv, 1208.0053 | | Services: | Forum | Review | PDF | Favorites | |
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