Metric Diophantine Approximation on Manifolds
Metric Diophantine Approximation on Manifolds
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DOI：
10.1112/jlms/s2-14.1.43
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发表时间：
1976-10
影响因子：
1.2
通讯作者：
R. Baker
中科院分区：
文献类型：
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作者：
R. Baker
我们说，点p=（/？l5.，/？..||英尺，9||）> aT 1/n对于所有整数q> 0（||X||表示从x到最近整数的距离）。我们说P是非常好的逼近，如果对于某个e> 0，max（110，311，.，11 ft，q ||）<q-1/n-8在整数q> 0中有无穷多个解。R”中勒贝格测度意义下的几乎所有点jS既不是差逼近的也不是好逼近的[1;第七章]。如果我们考虑R”中流形r上的几乎所有点P（在T上的测度意义下），就会出现一个更困难的问题。1964年，VGSprindzuk证明了K.马勒指出曲线上几乎所有的点
We say that the point p=(/? l5...,/?„) of Rn is badly approximate if there is a number c> 0 such that max (HMII,...,|| ft, 9||)> aT1/n for all integers q> 0 (|| x|| denotes distance from x to the nearest integer). We say that P is very well approximable if, for some e> 0, max (110,311,..., 11ft, q\\)< q-l/n-8 has infinitely many solutions in integers q> 0. Almost all points jS in the sense of Lebesgue measure in R" are neither badly approximable nor very well approximable [1; Ch. VII]. A more difficult question arises if we consider almost all points P on a manifold r in R"(in the sense of the measure on T). VG Sprindzuk proved in 1964 the conjecture of K. Mahler that almost all points on the curve