On shrinking targets for Z^m actions on tori

TY - JOUR

T1 - On shrinking targets for Z^m actions on tori

AU - Bugeaud, Yann

AU - Harrap, Stephen

AU - Kristensen, Simon

AU - Velani, Sanju

PY - 2010/7

Y1 - 2010/7

N2 - 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.

AB - 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.

KW - Number Theory

UR - http://www.scopus.com/inward/record.url?scp=78751628334&partnerID=8YFLogxK

U2 - 10.1112/S0025579310001130

DO - 10.1112/S0025579310001130

M3 - Article

SN - 2041-7942

VL - 56

SP - 193

EP - 202

JO - Mathematika

JF - Mathematika

IS - 2

ER -