This thesis is concerned with number-theoretic properties of real numbers, approached from two directions: the first is representation of real numbers via numeration systems and the (measure-theoretic) size of sets of real numbers with prescribed Diophantine approximation properties and the second is numbers defined by digit restrictions.
The results presented stem from one paper and two preprints, written in collaboration with Ying Wai Lee, Jaroslav Hančl, and Simon Kristensen, concerning missing-digits sets, expressible sets, and quantitative Khintchine-type theorems.
In the first part of the thesis we study missing-digits sets arising from base-$b$ expansions, continued fractions, and Lüroth expansions. We use their interpretations as Cantor sets to establish several results on decomposing a real number into a sum of numbers whose Lüroth expansion digits are restricted in particular ways. We also use the interpretation of missing-digits sets as Cantor sets to give a lower bound on the Hausdorff dimension of expressible sets.
In the second part, we turn to metric Diophantine approximation. We prove a quantitative version of the convergence part of Khintchine's theorem in metric Diophantine approximation, but where the approximated points lie on the standard parabola and the approximation function is not assumed to be monotonic. Also included in this section is a note on twisted inhomogeneous Dirichlet non-improvability
