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

TY - JOUR

T1 - On a problem in simultaneous Diophantine approximation

T2 - Schmidt's conjecture

AU - Badziahin, Dzmitry

AU - Pollington, Andrew

AU - Velani, Sanju

PY - 2011/11

Y1 - 2011/11

N2 - 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.

AB - 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.

KW - Number Theory

UR - http://www.scopus.com/inward/record.url?scp=80054984349&partnerID=8YFLogxK

U2 - 10.4007/annals.2011.174.3.9

DO - 10.4007/annals.2011.174.3.9

M3 - Article

VL - 174

SP - 1837

EP - 1883

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 3

ER -