Twisted inhomogeneous Diophantine approximation and badly approximable sets

TL;DR

This paper characterizes the set of badly approximable pairs in twisted inhomogeneous Diophantine approximation and shows that their twisted inhomogeneous analogues have full Hausdorff dimension, extending previous work for the case i=j=1/2.

Contribution

It introduces a new characterization of Bad(i, j) using well-approximable vectors and proves the full Hausdorff dimension of twisted analogues for pairs in Bad(i, j).

Findings

Characterization of Bad(i, j) via well-approximable vectors.

Full Hausdorff dimension of twisted Bad^x(i, j) for x in Bad(i, j).

Generalization of Kurzweil's work for i=j=1/2.

Abstract

For any real pair i, j geq 0 with i+j=1 let Bad(i, j) denote the set of (i, j)-badly approximable pairs. That is, Bad(i, j) consists of irrational vectors x:=(x_1, x_2) in R^2 for which there exists a positive constant c(x) such that max {||qx_1||^(-i), ||qx_2||^(-j)} > c(x)/q for all q in N. A new characterization of the set Bad(i, j) in terms of `well-approximable' vectors in the area of 'twisted' inhomogeneous Diophantine approximation is established. In addition, it is shown that Bad^x(i, j), the `twisted' inhomogeneous analogue of Bad(i, j), has full Hausdorff dimension 2 when x is chosen from the set Bad(i, j). The main results naturally generalise the i=j=1/2 work of Kurzweil.