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

Badziahin, Dzmitry; Pollington, Andrew; Velani, Sanju

doi:10.4007/annals.2011.174.3.9

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

Badziahin, Dzmitry; Pollington, Andrew; Velani, Sanju

Authors

Dzmitry Badziahin

Andrew Pollington

Sanju Velani

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.