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 dc.contributor.advisor Fraser, Jonathan dc.contributor.author Yu, Han dc.coverage.spatial vii, 101 p. en_US dc.date.accessioned 2019-07-24T15:04:04Z dc.date.available 2019-07-24T15:04:04Z dc.date.issued 2019-12-03 dc.identifier.uri http://hdl.handle.net/10023/18157 dc.description.abstract In the first part of this thesis we introduce a new dimension spectrum motivated by the Assouad dimension; a familiar notion of dimension which, for a given metric space, returns the minimal exponent α ≥ 0 such that for any pair of scales 0 < r < R, any ball of radius R may be covered by a constant times (R/r)ᵅ balls of radius r. To each 𝛩 ∈ (0,1), we associate the appropriate analogue of the Assouad dimension with the restriction that the two scales r and R used in the definition satisfy log R/log r = 𝛩. The resulting 'dimension spectrum' (as a function of 𝛩) thus gives finer geometric information regarding the scaling structure of the space and, in some precise sense, interpolates between the upper box dimension and the Assouad dimension. This latter point is particularly useful because the spectrum is generally better behaved than the Assouad dimension. We also consider the corresponding 'lower spectrum', motivated by the lower dimension, which acts as a dual to the Assouad spectrum. We conduct a detailed study of these dimension spectra; including analytic and geometric properties. We also compute the spectra explicitly for some common examples of fractals including decreasing sequences with decreasing gaps and spirals with sub-exponential and monotonic winding. We also give several applications of our results, including: dimension distortion estimates under bi-Hölder maps for Assouad dimension. We compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of 'tail densities' defined for subsets of the integers. en_US In the second part of this thesis, we study the Assouad dimension of sets of integers and deduce a weak solution to the Erdős-Turán conjecture. Let 𝐹 ⊂ ℕ. If $\sum_{n\in F}n^{-1}=\infty$ then 𝐹 "asymptotically" contains arbitrarily long arithmetic progressions. dc.language.iso en en_US dc.publisher University of St Andrews dc.rights Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International * dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/4.0/ * dc.subject Assouad type spectra en_US dc.subject Assouad dimension en_US dc.subject.lcc QA611.3Y8 dc.subject.lcsh Dimension theory (Topology) en dc.subject.lcsh Metric spaces en dc.subject.lcsh Fractals en dc.title Assouad type dimensions and dimension spectra en_US dc.type Thesis en_US dc.contributor.sponsor University of St Andrews. School of Mathematics and Statistics en_US dc.contributor.sponsor University of Manchester en_US dc.type.qualificationlevel Doctoral en_US dc.type.qualificationname PhD Doctor of Philosophy en_US dc.publisher.institution The University of St Andrews en_US