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Publication - Dr Demi Allen

    The Mass Transference Principle: Ten years on

    Citation

    Allen, D & Troscheit, S, 2019, ‘The Mass Transference Principle: Ten years on’. in: Erin Pearse, John Rock, Tony Samuel, Robert Niemeyer (eds) Horizons of Fractal Geometry and Complex Dimensions. American Mathematical Society, pp. 1-33

    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 (most cases of) a general inhomogeneous Jarnik–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.

    Full details in the University publications repository