A metrical theorem in geometry of numbers
A metrical theorem in geometry of numbers
复制标题
数几何中的度量定理
DOI：
10.1090/s0002-9947-1960-0117222-9
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发表时间：
1960
影响因子：
1.3
通讯作者：
W. Schmidt
中科院分区：
文献类型：
--
作者：
W. Schmidt
对于S中的格点的数目。在这里，以及在本文中，格点是具有整数坐标的点。如果S是有限体积V（S）的Borel集，人们会期望L（S）与V（S）具有相同的数量级。因此，我们定义“差异”D（S）为：（1）D（S）= I L（S）V（S）1作为L（S）的伴随，我们引入P（S），S中本原格点的数目。（如果一个格点的坐标是互质的，那么它就是本原的。）设（2）E（S）= I P（S）t（n）V（S）t ′.我|其次，设1'是一个有限体积的Borel集族，使得（i）如果SeC，TCb，则SOT或TCS。(ii)存在具有任意大V（S）的SeI 4。最后，在本文中，+1（s），s _ 0，应该是一个正的，非减的函数，使得fo_4（s）-1ds存在。定理1.假设n >2。则对几乎每一线性变换A（“几乎每一”是指A的矩阵表示所导出的n ~ 2维欧氏度量），（3）D（AS）= O（V-1/2 J ~ 2log V ~ ytI ~ 2（log V）），（4）E（AS）= O（V-1/2log V ~ 4，1/2（log V））.更明确地说，对于几乎每个A，都存在常数c1（A），c2（A），使得D（AS）c2（A）和SE+q。在R2中，我们的结果稍弱：1959年3月13日由编辑收到。516
for the number of lattice-points in S. Here, and throughout this paper, a lattice-point is a point with integral coordinates. If S is a Borel set of finite volume V(S), one would expect that L(S) is of about the same order of magnitude as V(S). Hence we define the "discrepancy" D(S) by (1) D(S) = I L(S)V(S)1 As a companion for L(S), we introduce P(S), the number of primitive lattice-points in S. (A lattice-point is primitive, if its coordinates are relatively prime.) We put (2) E(S) = I P(S)t(n)V(S)'. I| Next, let 1' be a family of Borel sets with finite volumes, such that (i) If SeC, TCb, then either SOT or TCS. (ii) There exist SeI4 with arbitrarily large V(S). Finally, throughout this paper, +1(s), s _ 0, should be a positive, nondecreasing function, such that fo4(s)-1ds exists. THEOREM 1. Suppose n >2. Then for almost every linear transformation A ("almost every" in the sense of the n2-dimensional euclidean metric induced by matrix-representation for A), (3) D(AS) = O(V-12J2 log V ytI2(log V)), (4) E(AS) = O(V-1/2 log V 4,1/2(log V)) for SC?. More explicitly, for almost every A there exist constants ci(A), c2(A), such that D(AS) c2(A) and SE+q. In R2 our results are a little weaker: Received by the editors March 13, 1959. 516