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

Badziahin, Dzmitry; Pollington, Andrew; Velani, Sanju

Abstract

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

Citation

Badziahin, D., Pollington, A., & Velani, S. (2011). On a problem in simultaneous Diophantine approximation: Schmidt's conjecture. Annals of Mathematics, 174(3), 1837-1883. https://doi.org/10.4007/annals.2011.174.3.9

• Journal Article Type: Article
• Publication Date: Nov 1, 2011
• Deposit Date: May 30, 2011
• Publicly Available Date: May 6, 2014
• Journal: Annals of Mathematics
• Print ISSN: 0003-486X
• Electronic ISSN: 1939-8980
• Publisher: Department of Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 174
• Issue: 3
• Pages: 1837-1883
• DOI: https://doi.org/10.4007/annals.2011.174.3.9
• Keywords: Cantor sets, Hausdorff dimension, Simultaneously badly approximable numbers.
• Public URL: https://durham-repository.worktribe.com/output/1531778