Badly approximable vectors on a vertical Cantor set

Abstract

For $i,j>0,i+j=1$, the set of badly approximable vectors with weight $(i,j)$ is defined by $Bad(i,j)=\{(x,y)\in {\mathbb{R}}^2:\exists c>0\forall q\in \mathbb{N},\max \{q||qx|{|}^{1/i},q||qy|{|}^{1/j}\}>c\}$, where $||x||$ is the distance of $x$ to the nearest integer. In 2010 Badziahin-Pollington-Velani solved Schmidt's conjecture which was stated in 1982, proving that $Bad(i,j)\cap Bad(j,i)$ is nonempty. Using Badziahin-Pollington-Velani's technique with reference to fractal sets, we were able to improve their results: Assume that we are given a sequence $(i_t,j_t)$ with $i_t,j_t>0,i_t+j_t=1$. Then, the intersection of $Bad(i_t,j_t)$ over all t is nonempty.

References in corpus (1)

• Badziahin-Pollington-Velani's theorem and Schmidt's game

Cited by in corpus (7)

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• Badziahin-Pollington-Velani's theorem and Schmidt's game
• Jarnik-type Inequalities
• Schmidt Games and Conditions on Resonant Sets
• Bad($\mathbf{w}$) is hyperplane absolute winning
• Quantitative non-divergence and Diophantine approximation on manifolds
• Badly approximable points on planar curves and a problem of Davenport