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

TY - UNPB

T1 - The Hausdorff measure version of Gallagher's theorem -- closing the gap and beyond

AU - Simmons, David Samuel

AU - Hussain, Mumtaz

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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.

AB - 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.

M3 - Preprint

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

ER -