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On a problem in simultaneous Diophantine approximation: Schmidt's conjecture
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| Journal | Annals of Mathematics |
|---|---|
| Date | Published - Nov 2011 |
| Issue number | 3 |
| Volume | 174 |
| Number of pages | 46 |
| Pages (from-to) | 1837-1883 |
| Original language | English |
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.
- Number Theory
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