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

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F19%3A00511319" target="_blank" >RIV/67985840:_____/19:00511319 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1007/s10474-019-00992-9" target="_blank" >http://dx.doi.org/10.1007/s10474-019-00992-9</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s10474-019-00992-9" target="_blank" >10.1007/s10474-019-00992-9</a>

Result language
angličtina
Original language name
The de Bruijn-Erdos theorem from a Hausdorff measure point of view
Original language description
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.
Czech name
—
Czech description
—

Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Similar results(10)