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dc.contributor.authorAdam-Day, B.
dc.contributor.authorAshcroft, C.
dc.contributor.authorOlsen, L.
dc.contributor.authorPinzani, N.
dc.contributor.authorRizzoli, A.
dc.contributor.authorRowe, J.
dc.date.accessioned2019-10-02T23:37:30Z
dc.date.available2019-10-02T23:37:30Z
dc.date.issued2018-12
dc.identifier.citationAdam-Day , B , Ashcroft , C , Olsen , L , Pinzani , N , Rizzoli , A & Rowe , J 2018 , ' On the average box dimensions of graphs of typical continuous functions ' , Acta Mathematica Hungarica , vol. 156 , no. 2 , pp. 263-302 . https://doi.org/10.1007/s10474-018-0871-2en
dc.identifier.issn0236-5294
dc.identifier.otherPURE: 256346414
dc.identifier.otherPURE UUID: c7a4ef4a-499c-49ff-8885-765cbbff02a4
dc.identifier.otherScopus: 85054542052
dc.identifier.otherORCID: /0000-0002-8353-044X/work/60630678
dc.identifier.otherWOS: 000449099200001
dc.identifier.urihttp://hdl.handle.net/10023/18603
dc.description.abstractLet X be a bounded subset of Rd and write Cu(X) for the set of uniformly continuous functions on X equipped with the uniform norm. The lower and upper box dimensions, denoted by dim̲B(graph(f)) and dim ¯ B(graph (f)) , of the graph graph (f) of a function f∈ Cu(X) are defined by dim̲B(graph(f))=limδ↘0inflogNδ(graph(f))-logδ,dim¯B(graph(f))=limδ↘0suplogNδ(graph(f))-logδ,where Nδ(graph (f)) denotes the number of δ-mesh cubes that intersect graph (f). Hyde et al. have recently proved that the box counting function(∗)logNδ(graph(f))-logδof the graph of a typical function f∈ Cu(X) diverges in the worst possible way as δ↘ 0. More precisely, Hyde et al. showed that for a typical function f∈ Cu(X) , the lower box dimension of the graph of f is as small as possible and if X has only finitely many isolated points, then the upper box dimension of the graph of f is as big as possible. In this paper we will prove that the box counting function (*) of the graph of a typical function f∈ Cu(X) is spectacularly more irregular than suggested by the result due to Hyde et al. Namely, we show the following surprising result: not only is the box counting function in (*) divergent as δ↘ 0 , but it is so irregular that it remains spectacularly divergent as δ↘ 0 even after being “averaged" or “smoothened out" using exceptionally powerful averaging methods including all higher order Hölder and Cesàro averages and all higher order Riesz–Hardy logarithmic averages. For example, if the box dimension of X exists, then we show that for a typical function f∈ Cu(X) , all the higher order lower Hölder and Cesàro averages of the box counting function (*) are as small as possible, namely, equal to the box dimension of X, and if, in addition, X has only finitely many isolated points, then all the higher order upper Hölder and Cesàro averages of the box counting function (*) are as big as possible, namely, equal to the box dimension of X plus 1.
dc.format.extent40
dc.language.isoeng
dc.relation.ispartofActa Mathematica Hungaricaen
dc.rightsCopyright © Akadémiai Kiadó, Budapest, Hungary 2018. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at: https://doi.org/10.1007/s10474-018-0871-2en
dc.subjectBaire categoryen
dc.subjectBox dimensionen
dc.subjectCesàro meanen
dc.subjectContinuous functionsen
dc.subjectHölder meanen
dc.subjectRiesz–Hardy meanen
dc.subjectQA Mathematicsen
dc.subjectT-NDASen
dc.subject.lccQAen
dc.titleOn the average box dimensions of graphs of typical continuous functionsen
dc.typeJournal articleen
dc.description.versionPostprinten
dc.contributor.institutionUniversity of St Andrews.Pure Mathematicsen
dc.identifier.doihttps://doi.org/10.1007/s10474-018-0871-2
dc.description.statusPeer revieweden
dc.date.embargoedUntil2019-10-03


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