Zafeiropoulos, Agamemnon (2017) Inhomogeneous Diophantine approximation, M_0 sets and projections of fractals. PhD thesis, University of York.
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Abstract
The thesis deals with two main subjects, one being metric Diophantine approximation and the other Fractal Geometry. As far as the first subject is concerned, the results presented lie in the setup of inhomogeneous Diophantine approximation. The following is shown: suppose $\ca{A}=(q_n)_{n=1}^{\infty}\subseteq \mathbb{N}$ is a lacunary sequence and $\mu$ is a probability measure with Fourier transform of a prescribed logarithmic decay rate. Then for any $\gamma\in\mathbb{R}$ and any decreasing approximating function $\psi:\mathbb{N}\rightarrow \mathbb{R}^+$, the set $W_{\ca{A}}(\gamma,\psi)= \{x \in [0,1) : \q_nx  \gamma\ \leq \psi(q_n) \text{\ for i.m. } n\in\mathbb{N} \}$ satisfies a Khintchinetype law with respect to the measure $\mu$. This result builds on the work of Pollington and Velani. It is also shown that $W(\gamma,\psi)$ is a Salem set, generalising a result of Kaufman. Regarding Fractal Geometry, we present a refinement of Marstrand's famous projection theorem for arbitrary dimension functions. We state an analogue of Kaufman's result on the dimension of the set of angles of exceptional projections and discuss on the necessity of the conditions imposed in our main theorem.
Item Type:  Thesis (PhD) 

Academic Units:  The University of York > Mathematics (York) 
Depositing User:  Mr Agamemnon Zafeiropoulos 
Date Deposited:  22 Nov 2017 13:37 
Last Modified:  22 Nov 2017 13:37 
URI:  http://etheses.whiterose.ac.uk/id/eprint/18705 
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 Inhomogeneous Diophantine approximation, M_0 sets and projections of fractals. (deposited 22 Nov 2017 13:37) [Currently Displayed]