Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T01:06:44.986Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal intersections of plane lattice points with vertical and horizontal straight lines

Published online by Cambridge University Press:  24 October 2008

Sui-Sun Cheng  and
Yu-Tai Huang
Sui-Sun Cheng
Affiliation:
Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan 300
Yu-Tai Huang
Affiliation:
Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan 300
Get access Rights & Permissions [Opens in a new window]

Extract

A point in the plane is said to be a lattice point if it has integral coordinates. Given a set C of lattice points and two non-negative integers m and n, we are concerned with determining a set of m vertical and a set of n horizontal straight lines so that the number of intersections of these sets is maximal. In general, this problem is rather difficult, since we have to make simultaneous decisions about the positions of the vertical and horizontal straight lines. Our purpose here is to study a set of lattice points of the form (i, j) such that i, j, i + j and i − j are bounded by fixed integers. We shall show that, in this case, sequential decisions can be made about the two families of straight lines.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 3 , November 1983 , pp. 381 - 388
Copyright
Copyright © Cambridge Philosophical Society 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities (Cambridge University Press, 1964).Google Scholar
[2] Gabriel, R. M.. The rearrangement of positive Fourier coefficients. Proc. London Math. Soc. (2) 33 (1932), 32–51.CrossRefGoogle Scholar