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Lower Bounds for the Square-to-Linear Ratio for Plane Peano Curves

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Abstract

It is proved that, for any map of the unit interval onto the unit square, there exist two points in the interval such that the squared Euclidean distance between their images exceeds the distance between them on the interval at least by a factor of \(3.625\). The additional condition that the images of the interval endpoints belong to opposite sides of the square increases this factor to more than \(4\).

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References

  1. E. V. Shchepin, “On Fractal Peano curves,” Proc. Steklov Inst. Math. 247, 272–280 (2004).

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 119991, Russia

    E. V. Shchepin

  2. Lomonosov Moscow State University, Moscow, 119991, Russia

    E. Yu. Mychka

Authors
  1. E. V. Shchepin
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  2. E. Yu. Mychka
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Correspondence to E. V. Shchepin.

Additional information

Translated from Matematicheskie Zametki, 2021, Vol. 110, pp. 289-296 https://doi.org/10.4213/mzm13069.

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Shchepin, E.V., Mychka, E.Y. Lower Bounds for the Square-to-Linear Ratio for Plane Peano Curves. Math Notes 110, 267–272 (2021). https://doi.org/10.1134/S0001434621070282

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  • DOI: https://doi.org/10.1134/S0001434621070282

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