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

Abstract

For any i, j ≥ 0 with i + j = 1, let Bad(i, j) denote the set of points (x, y) ∈ R{double-struck}2 for which max > c/q for all q ∈ (N{double-struck}). 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 approxi-mation.