A Note on the Number of General 4-holes in (Perturbed) Grids

• Published in: Discrete and Computational Geometry and Graphs Vol. 9943; pp. 1 - 12
• Main Authors: Aichholzer, O., Hackl, T., Valtr, P., Vogtenhuber, B.
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2016; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Algorithms & data structures; Collinear Point; Discrete mathematics; Empty Triangle; Graphics programming; Grid Point; Lattice Line; Prime Segment
• ISBN: 9783319485317; 3319485318
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-48532-4_1

Considering a variation of the classical Erdős-Szekeres type problems, we count the number of general 4-holes (not necessarily convex, empty 4-gons) in squared Horton sets of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{n}\!\times \!\sqrt{n}$$\end{document}. Improving on previous upper and lower bounds we show that this number is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varTheta (n^2\log n)$$\end{document}, which constitutes the currently best upper bound on minimizing the number of general 4-holes for any set of n points in the plane. To obtain the improved bounds, we prove a result of independent interest. We show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{d=1}^n \frac{\varphi (d)}{d^2} = \varTheta (\log n)$$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (d)$$\end{document} is Euler’s phi-function, the number of positive integers less than d which are relatively prime to d. This arithmetic function is also called Euler’s totient function and plays a role in number theory and cryptography.