Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

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JournalActa Arithmetica
DateIn preparation - 25 Jul 2015
DateAccepted/In press - 25 Jul 2016
DateE-pub ahead of print (current) - 22 Feb 2017
Issue number4
Volume177
Number of pages14
Early online date22/02/17
Original languageEnglish

Abstract

For any j_1,...,j_n>0 with j_1+...+j_n=1 and any x \in R^n, we consider the set of points y \in R^n for which max_{1\leq i\leq n}(||qx_i-y_i||^{1/j_i})>c/q for some positive constant c=c(y) and all q\in N. These sets are the `twisted' inhomogeneous analogue of Bad(j_1,...,j_n) in the theory of simultaneous Diophantine approximation. It has been shown that they have full Hausdorff dimension in the non-weighted setting, i.e provided that j_i=1/n, and in the weighted setting when x is chosen from Bad(j_1,...,j_n). We generalise these results proving the full Hausdorff dimension in the weighted setting without any condition on x.

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© Instytut Matematyczny PAN, 2017. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

    Research areas

  • math.NT

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