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dc.contributor.advisorAzzam, Jonas
dc.contributor.advisorGimperlein, Heiko
dc.contributor.authorHyde, Matthew
dc.date.accessioned2021-12-10T15:29:58Z
dc.date.available2021-12-10T15:29:58Z
dc.date.issued2021-11-27
dc.identifier.urihttps://hdl.handle.net/1842/38339
dc.identifier.urihttp://dx.doi.org/10.7488/era/1604
dc.description.abstractIn this thesis, we discuss recent progress on higher dimensional analogues to the Analyst’s Travelling Salesman Theorem (TST) of Peter Jones. The TST characterizes subsets of rectifiable curves in the plane, via a multiscale sum of β-numbers. These β-numbers measure how far a set E deviates from a straight line at a particular scale and location. This idea was extended by Okikiolu to subsets of Euclidean space and by Schul to subsets of Hilbert space. In 2018, Azzam and Schul introduced a variant of the Jones β-number. With this, they, and separately Villa, proved similar results for higher dimensional subsets of Euclidean space. In particular, Villa characterizes the lower regular subsets of a certain class of d-dimensional surfaces, first introduced in a 2004 paper of David. We prove an analogous result for arbitrary subsets of Euclidean space, we do not need to assume lower regularity. To do this, we introduce a new d-dimensional variant of the Jones β-number that is defined for any set in Euclidean space. A significant portion of this thesis is dedicated to studying the properties of this β-number.en
dc.description.abstract2022-11-27en
dc.contributor.sponsorEngineering and Physical Sciences Research Council (EPSRC)en
dc.language.isoenen
dc.publisherThe University of Edinburghen
dc.relation.hasversionMatthew Hyde. An analyst’s travelling salesman theorem for general sets in euclidean space. arXiv preprint arXiv:2006.16677, 2020.en
dc.subjectgeometric object propertiesen
dc.subjectmulti-scale flatnessen
dc.subjectMandelbroten
dc.subjectAnalyst’s Travelling Salesman theoremen
dc.subjectrectifiable curves in the planeen
dc.subjectEuclidean spaceen
dc.subjectHilbert spaceen
dc.subjectd-dimensional surfacesen
dc.titleA d-dimensional analyst's Travelling Salesman theorem for arbitrary sets in Euclidean spaceen
dc.typeThesis or Dissertationen
dc.type.qualificationlevelDoctoralen
dc.type.qualificationnamePhD Doctor of Philosophyen
dc.rights.embargodate2022-11-27en
dcterms.accessRightsRestricted Accessen


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