Weighted simultaneous Diophantine approximation in a variety of settings

This thesis considers weighted simultaneous Diophantine approximation in a variety of settings, including approximation over real manifolds, p-adic manifolds and p-adic coordinate hyperplanes. In each of these lower bounds on the Hausdorff dimension are obtained via appropriate Mass Transference Principle theorems. Weighted simultaneous approximation sets are often described by limsup sets of rectangles, so Mass Transference Principles on rectangles are favoured. Examples of these include the Mass Transference Principle from balls to rectangles (Wang, Wu, Xu, 2015), and the Mass Transference Principle from rectangles to rectangles (Wang, Wu 2021). Chapters 1 and 2 provide an introduction to real and p-adic Diophantine approximation. Chapter 3 introduces the Mass Transference Principle, given by Beresnevich and Velani in 2006, and recent variations. These Theorems are vital in the proofs of results in later chapters. In Chapter 4, Diophantine approximation over manifolds is introduced and a survey of recent results is given. It the latter part of the chapter a Dirichlet style Theorem for approximable points over manifolds is proven, which generalises a similar result in (Beresnevich, Lee, Vaughan, Velani 2017). Such result allows us to apply a Mass Transference Principle result and obtain a lower bound on the Hausdorff dimension of weighted approximable points over manifolds. In Chapter 5, a variety of results in p-adic weighted Diophantine approximation are proven. Furthermore, a similar result to that established in Chapter 4 is proven for p-adic approximable points over manifolds. In Chapter 6 the Hausdorff dimension of p-adic approximable points over coordinate hyperplanes is proven. The result relies on a count for the number rational approximations to a p-adic integer, which is proven using p-adic approximation lattices. The thesis is concluded by providing a brief survey on S-arithmetic Diophantine approximation. This is followed by a discussion on how results found throughout the thesis could be replicated in the S-arithmetic setting.