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On Simultaneous Diophantine Approximation in the Vector Space Q+Qα

✍ Scribed by Edward B Burger

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
135 KB
Volume
82
Category
Article
ISSN
0022-314X

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✦ Synopsis

Beginning with an improvement to Dirichlet's Theorem on simultaneous approximation, in this paper we introduce an alogrithm, in sympathy with the classical continued fraction algorithm, to generate the sequence of best approximates to the system max[&: 0 n&, &: 1 n&, ..., &: L n&] in the case when : 0 , : 1 , ..., : L are elements of the vector space Q+Q:. We then produce best possible upper bounds for the associated diophantine inequality. In addition, we consider implications within the geometry of numbers and investigate the converse of the sharpened version of Dirichlet's result. Finally, we close with some remarks on the Littlewood Conjecture in this context.

2000 Academic Press

We remark that the exponents 1Â(L+1) in the previous inequalities cannot be improved for badly approximable (L+1)-tuples, that is, for those

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