An improved point‐line incidence bound over arbitrary fields

@article{Stevens2017AnIP,
  title={An improved point‐line incidence bound over arbitrary fields},
  author={S. Stevens and F. D. Zeeuw},
  journal={Bulletin of The London Mathematical Society},
  year={2017},
  volume={49},
  pages={842-858}
}
We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field F, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that m points and n lines in F2, with m7/8<n<m8/7, determine at most O(m11/15n11/15) incidences (where, if F has positive characteristic p, we assume m−2n13≪p15). This improves on the previous best-known bound, due to Jones. To obtain our bound, we first prove an optimal point-line incidence bound on… Expand
A short proof of Rudnev's point-plane incidence bound
A Second Wave of Expanders over Finite Fields
Improved Bounds for Pencils of Lines
INCIDENCES AND EXTREMAL PROBLEMS ON FINITE POINT SETS
Bisector energy and pinned distances in positive characteristic.
Upper and lower bounds for rich lines in grids
Extending Erdős-Beck's theorem to higher dimensions
  • Thao T. Do
  • Mathematics, Computer Science
  • Comput. Geom.
  • 2020
New bounds for distance-type problems over prime fields
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 52 REFERENCES
Incidences in three dimensions and distinct distances in the plane
On the Number of Incidences Between Points and Planes in Three Dimensions
  • M. Rudnev
  • Mathematics, Computer Science
  • Comb.
  • 2018
Further improvements to incidence and Beck-type bounds over prime finite fields
On the Erdos distinct distance problem in the plane
Incidence Theorems and Their Applications
  • Zeev Dvir
  • Mathematics, Computer Science
  • Found. Trends Theor. Comput. Sci.
  • 2012
Extremal problems in discrete geometry
Areas of triangles and Beck’s theorem in planes over finite fields
An improved incidence bound for fields of prime order
...
1
2
3
4
5
...