Number of the records: 1
The de Bruijn-Erdos theorem from a Hausdorff measure point of view
- 1.
SYSNO ASEP 0511319 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title The de Bruijn-Erdos theorem from a Hausdorff measure point of view Author(s) Doležal, Martin (MU-W) RID, SAI, ORCID
Mitsis, T. (GR)
Pelekis, Christos (MU-W) SAI, RIDSource Title Acta Mathematica Hungarica. - : Springer - ISSN 0236-5294
Roč. 159, č. 2 (2019), s. 400-413Number of pages 14 s. Language eng - English Country HU - Hungary Keywords de Bruijn–Erdős theorem ; Hausdorff measure ; devil’s staircase ; piecewise monotone function Subject RIV BA - General Mathematics OECD category Pure mathematics R&D Projects GJ18-01472Y GA ČR - Czech Science Foundation (CSF) GA17-27844S GA ČR - Czech Science Foundation (CSF) Method of publishing Limited access Institutional support MU-W - RVO:67985840 UT WOS 000501828900004 EID SCOPUS 85074095413 DOI 10.1007/s10474-019-00992-9 Annotation 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. Workplace Mathematical Institute Contact Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Year of Publishing 2020 Electronic address http://dx.doi.org/10.1007/s10474-019-00992-9
Number of the records: 1