Title: The Mass Transference Principle: Ten Years On,
Abstract: In this article we discuss the Mass Transference Principle due to Beresnevich
and Velani and survey several generalisations and variants, both deterministic
and random. Using a Hausdorff measure analogue of the inhomogeneous
Khintchine-Groshev Theorem, proved recently via an extension of the Mass
Transference Principle to systems of linear forms, we give an alternative proof
of a general inhomogeneous Jarn\'{\i}k-Besicovitch Theorem which was originally
proved by Levesley. We additionally show that without monotonicity Levesley's
theorem no longer holds in general. Thereafter, we discuss recent advances by
Wang, Wu and Xu towards mass transference principles where one transitions from
$\limsup$ sets defined by balls to $\limsup$ sets defined by rectangles (rather
than from "balls to balls" as is the case in the original Mass Transference
Principle). Furthermore, we consider mass transference principles for
transitioning from rectangles to rectangles and extend known results using a
slicing technique. We end this article with a brief survey of random analogues
of the Mass Transference Principle.