On the theory of diophantine approximations. II (inhomogeneous problems)
On the theory of diophantine approximations. II (inhomogeneous problems)
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关于丢番图近似理论。
DOI：
10.1007/bf02023874
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发表时间：
1958
期刊：
影响因子：
--
通讯作者：
V. Sós
中科院分区：
文献类型：
--
作者：
V. Sós
用连分式算法统一处理了丢番图逼近的一维齐次问题。可以给出连分式的收敛和分母(Nebennenner)的几何解释，这种解释可以推广到非齐次情况，因此它提供了对这些情况的并行处理。这样，我们就可以证明一些关于非齐次情形的Borel型的简单定理，得到由c=inf sup inf xlax-y I.a~x>；O，y所定义的Khintchine常数c的新的上下界。JWS Cassel[1]用算术方法给出了一个类似于我们在本文中给出的非齐次情形的算法，R.Descomes[2]也使用了该算法。在脚注中对这两种方法进行了比较。在文[2]中，我们给出了无理a的连分式ns的几何解释和非齐次情形的相应算法，给出了与连分式的收敛和分母序列相对应的乘数序列S，(i：r)，进而给出了与非齐次情形相对应的乘数对序列Qj，(r Qf~(5))，它是S~(R)的一个子序列。我们称这些定理为Borel型，因为Borel改进了Hurwitz定理，使不等式
The one-dimensional homogeneous problems of diophantine approximations have a unified treatment by the algorithm of continued fractions. It is possible to give such a geometrical interpretation of convergents and by-denominators (Nebennenner) of continued fractions, which can be extended for the inhomogeneous case, and so it furnishes a parallel treatment of these cases. In this way, eg, it is possible to prove some simple theorems of Borel type for the inhomogeneous case, to get new lower and upper bounds for the Khintchine constant c defined by c= inf sup inf xlax--~--y I. a~ x> O, y integersThis last result will be treated in [5] and [6]. A similar algorithm, as we give in this paper for the inhomogeneous, case, is given in an arithmetical way by JWS CASSELS [1] and used also by R. DESCOMBES [2]. A comparison of both treatments is made in footnoteS. In {} 2 we give this geometrical interpretation of continued fractio, ns for an irrational a and the corresponding algorithm for the inhomogeneous case giving a sequence of multipla s,,(i: r which corresponds to the sequence of convergents and by-denominators of continued fractions, and further a sequence of pairs of multipla qj,,(r qf~(5) which is a subsequence of s~(r and corresponds to the sequence of convergents qj~ of a. In {} 3 we give the proof of some simple theorems of Borel type corresponding to the inhomogeneous case. We call these theorems Borel type, since BOReL sharpened HURWITZ'S theorem to the effect that the inequality