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

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Discrete and Computational Geometry and Graphs (JCDCGG 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9943))

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Abstract

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 \(\sqrt{n}\!\times \!\sqrt{n}\). Improving on previous upper and lower bounds we show that this number is \(\varTheta (n^2\log n)\), 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 \(\sum _{d=1}^n \frac{\varphi (d)}{d^2} = \varTheta (\log n)\), where \(\varphi (d)\) 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.

This work is partially supported by FWF projects I648-N18 and P23629-N18, by the OEAD project CZ 18/2015, and by the project CE-ITI no. P202/12/G061 of the Czech Science Foundation GAČR, and by the project no. 7AMB15A T023 of the Ministry of Education of the Czech Republic.

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Notes

  1. 1.

    If \(0<p<q\) are two coprime integers then there is a unique r, \(0<r<q\), such that \(pr \equiv 1 \mod q\).

References

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Author information

Authors and Affiliations

  1. Institute for Software Technology, Graz University of Technology, Graz, Austria

    O. Aichholzer, T. Hackl & B. Vogtenhuber

  2. Department of Applied Mathematics and Institute for Computer Science (ITI), Charles University, Prague, Czech Republic

    P. Valtr

Authors
  1. O. Aichholzer
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  2. T. Hackl
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  3. P. Valtr
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  4. B. Vogtenhuber
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Corresponding author

Correspondence to B. Vogtenhuber .

Editor information

Editors and Affiliations

  1. Tokyo University of Science , Tokyo, Japan

    Jin Akiyama

  2. The University of Electro-Communications , Tokyo, Japan

    Hiro Ito

  3. Tokai University , Tokyo, Japan

    Toshinori Sakai

  4. Osaka Prefecture University , Sakai, Japan

    Yushi Uno

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Aichholzer, O., Hackl, T., Valtr, P., Vogtenhuber, B. (2016). A Note on the Number of General 4-holes in (Perturbed) Grids. In: Akiyama, J., Ito, H., Sakai, T., Uno, Y. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2015. Lecture Notes in Computer Science(), vol 9943. Springer, Cham. https://doi.org/10.1007/978-3-319-48532-4_1

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  • DOI: https://doi.org/10.1007/978-3-319-48532-4_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-48531-7

  • Online ISBN: 978-3-319-48532-4

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