Modified Schmidt games and Diophantine approximation with weights

Abstract

We show that the sets of weighted badly approximable vectors in ${\mathbb{R}}^n$ are winning sets of certain games, which are modifications of $(\alpha ,\beta )$-games introduced by W. Schmidt in 1966. The latter winning property is stable with respect to countable intersections, and is shown to imply full Hausdorff dimension.

26 pages; a substantially revised version

References in corpus (1)

• Schmidt's Game on Certain Fractals

Cited by in corpus (3)

• Badly approximable systems of affine forms, fractals, and Schmidt games
• On a problem in simultaneous Diophantine approximation: Schmidt's conjecture
• Schmidt's game, fractals, and numbers normal to no base