Metric Diophantine approximation on manifolds by algebraic points

This thesis is concerned with various aspects of the metric theory of Diophantine Approximation by algebraic points. It is comprised of three introductory chapters, the presentation of our original work (Section 3.1, Chapters 4 and 5), and two appendices. At the end of Chapter 3 we introduce a simple application of the quantitative non-divergence estimates of Kleinbock and Margulis to a problem of approximation of points on a circle in the complex plane by ratios of Gaussian integers, which is motivated by recent advances in the theory of Wireless Communications. Then, in Chapter 4 we prove some partial results towards a Hausdorff measure description of the set of real numbers close to the zeros of polynomials of bounded degree, expanding on previous work of Hussain and Huang (see Section 1.2.1). Specifically, we use an estimate on the number of cubic polynomials with bounded discriminant due to Kaliada, Götze and Kukso and a measure bound due to Beresnevich to prove a convergence result for irreducible cubic polynomials, as well as for polynomials of arbitrary degree and large discriminant. Finally, in Chapter 5 we apply a quantitative non-divergence estimate and the theory of ubiquitous systems to derive both a counting lower bound and a divergence Hausdorff measure result for a problem of approximation on manifolds by points with algebraic conjugate coordinates, subject to a geometric constraint.