文摘
In Sharir and Solomon (2015), Sharir and Solomon showed that the number of incidences between mm distinct points and nn distinct lines in R4R4 is equation(1)O∗(m2∕5n4∕5+m1∕2n1∕2q1∕4+m2∕3n1∕3s1∕3+m+n),O∗m2∕5n4∕5+m1∕2n1∕2q1∕4+m2∕3n1∕3s1∕3+m+n,provided that no 2-flat contains more than ss lines, and no hyperplane or quadric contains more than qq lines, where the O∗O∗ hides a multiplicative factor of 2clogm for some absolute constant cc.In this paper we prove that, for integers m,nm,n satisfying n9∕8<m<n3∕2n9∕8<m<n3∕2, there exist mm points and nn lines on the quadratic hypersurface in R4R4{(x1,x2,x3,x4)∈R4∣x1=x22+x32−x42},such that (i) at most s=O(1)s=O(1) lines lie on any 2-flat, (ii) at most q=O(n∕m1∕3)q=O(n∕m1∕3) lines lie on any hyperplane, and (iii) the number of incidences between the points and the lines is Θ(m2∕3n1∕2)Θ(m2∕3n1∕2), which is asymptotically larger than the upper bound in (1), when n9∕8<m<n3∕2n9∕8<m<n3∕2. This shows that the assumption that no quadric contains more than qq lines (in the above mentioned theorem of Sharir and Solomon (2015)) is necessary in this regime of mm and nn.By a suitable projection from this quadratic hypersurface onto R3R3, we obtain mm points and nn lines in R3R3, with at most s=O(1)s=O(1) lines on a common plane, such that the number of incidences between the mm points and the nn lines is Θ(m2∕3n1∕2)Θ(m2∕3n1∕2). It remains an interesting question to determine if this bound is also tight in general.
