The de Bruijn-Erdos theorem from a Hausdorff measure point of view

0511319 - MÚ 2020 RIV HU eng J - Článek v odborném periodiku
Doležal, Martin - Mitsis, T. - Pelekis, Christos
The de Bruijn-Erdos theorem from a Hausdorff measure point of view.
Acta Mathematica Hungarica. Roč. 159, č. 2 (2019), s. 400-413. ISSN 0236-5294. E-ISSN 1588-2632
Grant CEP: GA ČR(CZ) GJ18-01472Y; GA ČR(CZ) GA17-27844S
Institucionální podpora: RVO:67985840
Klíčová slova: de Bruijn–Erdős theorem * Hausdorff measure * devil’s staircase * piecewise monotone function
Obor OECD: Pure mathematics
Impakt faktor: 0.588, rok: 2019
Způsob publikování: Omezený přístup
http://dx.doi.org/10.1007/s10474-019-00992-9

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= { (x, f1(x) , … , fn - 2(x) , α) : x∈ [0 , 1] } , where α is a fixed real number in [0,1] and f1, … , fn - 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 fi, i= 1 , … , 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.
Trvalý link: http://hdl.handle.net/11104/0301615