The Hausdorff measure version of Gallagher's theorem -- closing the gap and beyond

David Samuel Simmons, Mumtaz Hussain

Research output: Working paperPreprint


In this paper we prove an upper bound on the "size" of the set of multiplicatively ψ-approximable points in R^d for d>1 in terms of f-dimensional Hausdorff measure. This upper bound exactly complements the known lower bound, providing a "zero-full" law which relates the Hausdorff measure to the convergence/divergence of a certain series in both the homogeneous and inhomogeneous settings. This zero-full law resolves a question posed by Beresnevich and Velani (2015) regarding the "log factor" discrepancy in the convergent/divergent sum conditions of their theorem. We further prove the analogous result for the multiplicative doubly metric setup.
Original languageEnglish
Number of pages10
Publication statusSubmitted - 1 Dec 2016

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