On a problem in simultaneous Diophantine approximation: Schmidt's conjecture

Abstract

For any $i,j\ge 0$ with $i+j=1$, let $\bad(i,j)$ denote the set of points $(x,y)\in {\mathbb{R}}^2$ for which $\max \{\Vert qx{\Vert }^{1/i},\Vert qy{\Vert }^{1/j}\}>c/q$ for all $q\in \mathbb{N}$. Here $c=c(x,y)$ is a positive constant. Our main result implies that any finite intersection of such sets has full dimension. This settles a conjecture of Wolfgang M. Schmidt in the theory of simultaneous Diophantine approximation.

43 pages, 1 figure. A relatively minor mistake at the beginning of Section 4 (Proof of Theorem 3) that deals with the situation of parallel lines is corrected.

References in corpus (1)

• Modified Schmidt games and Diophantine approximation with weights

Cited by in corpus (3)

• Badly approximable points on manifolds
• The mixed Schmidt conjecture in the theory of Diophantine approximation
• On certain Littlewood-like and Schmidt-like problems in inhomogeneous Diophantine approximations