Twisted inhomogeneous Diophantine approximation and badly approximable sets
arXiv:1003.2362 · doi:10.4064/aa151-1-5
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.
arXiv comment32 pages. Updated 19/05/2010 (Revised definition on an "irrational" vector in introduction). Updated 21/06/2010 (Updated some references and updated some body text and abstract). Updated 17/05/2011 upon acceptance to be published in Acta Arithmetica (References, typing errors and some arguements updated to comply with referees suggestions. Some notation, descriptions also altered)