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A NOTE ON MATRIX APPROXIMATION IN THE THEORY OF MULTIPLICATIVE DIOPHANTINE APPROXIMATION

  • YUAN ZHANG (a1)

Abstract

We prove the Hausdorff measure version of the matrix form of Gallagher’s theorem in the inhomogeneous setting, thereby proving a conjecture posed by Hussain and Simmons [‘The Hausdorff measure version of Gallagher’s theorem—closing the gap and beyond’, J. Number Theory186 (2018), 211–225].

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[1] Allen, D. and Beresnevich, V., ‘A mass transference principle for systems of linear forms and its applications’, Compos. Math. 154(5) (2018), 10141047.
[2] Beresnevich, V., Haynes, A. and Velani, S., ‘Sums of reciprocals of fractional parts and multiplicative Diophantine approximation’, Mem. Amer. Math. Soc. to appear.
[3] Beresnevich, V. and Velani, S., ‘A note on three problems in metric Diophantine approximation’, in: Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211229.
[4] Chow, S., ‘Bohr sets and multiplicative diophantine approximation’, Duke Math. J. 167 (2018), 16231642.
[5] Gallagher, P., ‘Metric simultaneous diophantine approximation’, J. Lond. Math. Soc. 37 (1962), 387390.
[6] Hussain, M. and Simmons, D., ‘The Hausdorff measure version of Gallagher’s theorem—closing the gap and beyond’, J. Number Theory 186 (2018), 211225.
[7] Mattila, P., Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, 44 (Cambridge University Press, Cambridge, 1995).
[8] Sprindžuk, V., Metric Theory of Diophantine Approximations, Translations of Mathematical Monographs, 25 (American Mathematical Society, Providence, RI, 1969), translated from the Russian by B. Volkmann.
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A NOTE ON MATRIX APPROXIMATION IN THE THEORY OF MULTIPLICATIVE DIOPHANTINE APPROXIMATION

  • YUAN ZHANG (a1)

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