# Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

Paloma Bengoechea, Nikolay Moshchevitin

Research output: Contribution to journalArticlepeer-review

## 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.
Original language English 14 Acta Arithmetica 177 4 22 Feb 2017 https://doi.org/10.4064/aa8234-11-2016 E-pub ahead of print - 22 Feb 2017

### Bibliographical note

© Instytut Matematyczny PAN, 2017. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

## Keywords

• math.NT
• ### Programme Grant-New Frameworks in metric Number Theory

EPSRC

1/06/1230/11/18

Project: Research project (funded)Research