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ON SHRINKING TARGETS FOR ℤ m ACTIONS ON TORI
Author(s) -
Bugeaud Yann,
Harrap Stephen,
Kristensen Simon,
Velani Sanju
Publication year - 2010
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579310001130
Subject(s) - mathematics , combinatorics , hausdorff dimension , intersection (aeronautics) , irrational number , torus , constant (computer programming) , dimension (graph theory) , hausdorff space , point (geometry) , set (abstract data type) , geometry , discrete mathematics , computer science , engineering , programming language , aerospace engineering
Let A be an n × m matrix with real entries. Consider the set Bad A of x ∈[0,1) n for which there exists a constant c ( x )>0 such that for any q ∈ℤ m the distance between x and the point { A q } is at least c ( x )| q | − m / n . It is shown that the intersection of Bad A 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.