Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

TL;DR

This paper extends the understanding of the Hausdorff dimension of twisted Diophantine approximation sets, showing they have full dimension in weighted cases without restrictions on the point x.

Contribution

It generalizes previous results by proving full Hausdorff dimension in the weighted setting without conditions on x, broadening the scope of known properties.

Findings

Sets have full Hausdorff dimension in the weighted setting.

No restrictions on x are needed for the full dimension result.

Generalizes previous non-weighted and restricted weighted cases.

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.