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

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

2010-01-15

arxiv.org/abs/1001.2694v2

Mathematics

Number Theory

43 pages, 1 figure. A relatively minor mistake at the beginning of Section 4 (Proof of Theorem 3) that deals with the situatio

Scientific paper

For any $i,j \ge 0$ with $i+j =1$, let $\bad(i,j)$ denote the set of points $(x,y) \in \R^2$ for which $ \max \{\|qx\|^{1/i}, \|qy\|^{1/j} \} > c/q $ for all $ q \in \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.

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