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On shrinking targets for Z^m actions on tori

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JournalMathematika
DatePublished - Jul 2010
Issue number2
Volume56
Number of pages10
Pages (from-to)193-202
Original languageEnglish

Abstract

Let A be an n×m matrix with real entries. Consider the set BadA of x[0,1)n for which there exists a constant c(x)>0 such that for any qm the distance between x and the point {A q} is at least c(x)|q|-m/n. It is shown that the intersection of BadA with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

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  • Number Theory

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