The de Bruijn–Erdős theorem from a Hausdorff measure point of view

Abstract

Motivated by a well-known result in extremal set theory, due to Nicolaas Govert de Bruijn and Paul Erdős, we consider curves in the unit n-cube \([0,1]^n\) of the form

$$A= \bigl\{(x,f_1(x),\ldots,f_{n-2}(x),\alpha): x\in [0,1] \bigr\},$$

where \(\alpha\) is a fixed real number in [0,1] and \(f_1, \ldots, f_{n-2}\) are injective measurable functions from [0,1] to [0,1]. We refer to such a curve A as an n-de Bruijn–Erdős-set. Under the additional assumption that all functions \(f_i, i=1,\ldots,n-2,\) are piecewise monotone, we show that the Hausdorff dimension of A is at most 1 as well as that its 1-dimensional Hausdorff measure is at most n-1. Moreover, via a walk along devil’s staircases, we construct a piecewise monotone n-de Bruijn–Erdős-set whose 1-dimensional Hausdorff measure equals n-1.

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Acknowledgement

We are grateful to the anonymous referee for bringing to our attention references [6] and [9].

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Correspondence to Ch. Pelekis.

Additional information

Research of Doležal was supported by the GAČR project 17-27844S and RVO: 67985840.

Research of Pelekis was supported by the GAČR project 18-01472Y and RVO: 67985840.

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Doležal, M., Mitsis, T. & Pelekis, C. The de Bruijn–Erdős theorem from a Hausdorff measure point of view. Acta Math. Hungar. 159, 400–413 (2019). https://doi.org/10.1007/s10474-019-00992-9

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Key words and phrases

  • de Bruijn–Erdős theorem
  • Hausdorff measure
  • devil’s staircase
  • piecewise monotone function

Mathematics Subject Classification

  • 05D05
  • 28A78
  • 26A30