Badly approximable points in twisted Diophantine approximation and Hausdorff dimension
Number Theory
2016-07-26 v2
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.
Cite
@article{arxiv.1507.07119,
title = {Badly approximable points in twisted Diophantine approximation and Hausdorff dimension},
author = {Paloma Bengoechea and Nikolay Moshchevitin},
journal= {arXiv preprint arXiv:1507.07119},
year = {2016}
}