ON SHRINKING TARGETS FOR ℤ ^m ACTIONS ON TORI

A solution to a problem of Cassels and Diophantine properties of cubic numbers

“…The set y ∈ π −1 (x) : µ(y) > 0 (where x ∈ X 2 is arbitrary) has also been investigated, and it is known to have full Hausdorff dimension as well as being a winning set for Schmidt's game. See for example [13], [5], [22], and [11].…”

Сингулярные Диофантовы Системы А. Я. Хинчина И Их Применение

“…Простое доказательство это-го результата дали Я. Бюжо, С. Харрап, С. Кристенсен и С. Велани в [50]. Более того, они получили более сильную теорему, которую мы сейчас и сфор-мулируем.…”

A Note on Weighted Badly Approximable Linear Forms

“…Recently, Einsiedler & Tseng [8] extended the results of [4] and [22] to show amongst other related results that the set Bad L (n, m) is winning for any matrix L ∈ Mat n×m (R) (see also [12] and [16]). However, it appears their method cannot be extended to the weighted setting of the sets Bad L (k, n, m).…”

“…The set Bad {ur} was first shown in [4] to have full Hausdorff dimension for any sequence {u r } ∞ r=0 of non-zero integral vectors whose Euclidean norms form a lacunary sequence. We remark that in the case where the matrix M is not assumed to be chosen from Bad * (k, m, n) one may partition the sequence {u r } ∞ r=0 into a finite collection of subsequences {u t 0 +tr } ∞ r=0 such that each subsequence is lacunary with respect to the norm in (2.3); that is, the norm given by…”

“…In 2010 it was shown by Bugeaud et al [4] that this set, and its natural generalisation to higher dimensions, is of maximal Hausdorff dimension. † AND NIKOLAY MOSHCHEVITIN * Shortly afterwards, Tseng [22] demonstrated that Bad x enjoys the stronger property of being winning (in the sense of Schmidt 1 ) for all real numbers x.…”