Lines, line-point incidences and crossing families in dense sets

Abstract

LetP be a set ofn points in the plane. We say thatP isdense if the ratio between the maximum and the minimum distance inP is of order$O(\sqrt n )$. A setC of line segments in the plane is calleda crossing family if the relative interiors of any two line segments ofC intersect. Vertices of line segments of a crossing familyC are calledvertices of C. It is known that for any setP ofn points in general position in the plane there is a crossing family of size$\Omega (\sqrt n )$ with vertices inP. In this paper we show that ifP is dense then there is a crossing family of almost linear size with vertices inP.

The above result is related to well-known results of Beck and of Szemerédi and Trotter. Beck proved that any setP ofn points in the plane, not most of them on a line, determines at least Ω(n ²) different line. Szemerédi and Trotter proved that ifP is a set ofn points and ℒ is a set ofm lines then there are at mostO(m ^2/3 n ^2/3 +m+n) incidences between points ofP and lines of ℒ. We study whether or not the bounds shown by Beck and by Szemerédi and Trotter hold for any dense setP even if the notion of incidence is extended so that a point is considered to be incident to a linel if it lies in a small neighborhood ofl. In the first case we get very close to the conjectured bound Ω(n ²). In the second case we obtain a bound of order$O(\min \{ m^{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-\nulldelimiterspace} 4}} n^{{5 \mathord{\left/ {\vphantom {5 8}} \right. \kern-\nulldelimiterspace} 8}} \log ^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} n, n\sqrt m \} )$.