Distinct distances between lattice points
Distinct distances between lattice points
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格点之间的距离不同
DOI：
10.5169/seals-27359
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发表时间：
1970
期刊：
影响因子：
1.1
通讯作者：
P. Erdös
中科院分区：
文献类型：
--
作者：
R. Guy;P. Erdös
有多少个点（xi，yi），1 ~< i < k，整数坐标0 < xi，yi < n，可以选择所有相互距离不同？通过计算这样的距离和坐标差对，我们有~2/（n 2 l I-1，（1）/，使得k < n，并且对于2 < n < 7，可以获得这样的界限;例如，然而，数可以用不止一种方式表示为两个平方和的事实表明，当n > 15时，不能达到这个界限。朗道[4]的一个结果表明，小于x的整数的个数可表示为两个平方和，其渐近性为c1 x（logx）-12，因此我们可以用c2 n2（logn）-12来代替（1）的右成员，我们得到了上界，但它缺乏说服力，因为一维中相应的论证给出了错误的结果。另一方面，我们可以证明，对于任何e > 0和足够大的n，k > 1 i 2/3-E（4），通过下面的构造。依次选点;当k个点被选择时，取另一个点，使得（a）它不位于任何圆上，留下k个点中的一个点作为中心，并且由这些点确定的Ck 2个不同距离之一作为半径。(b)它与前k个点中的任何一个都不形成斜率为bla，（a，B）= 1的直线，其中b < n 1 / 3，B < 0 13。注意，特别地，没有两个点确定小于n 1/3的距离。
How many points (x i , y i), 1 ~< i < k, with integer coordinates 0 < xi, yi < n, may be chosen with all mutual distances distinct? Bv_ counting such distances, and pairs of differences of coordinates, we have ~2/ (n 2 l I-1, (1) / so that k < n, and for 2 < n < 7 such a bound can be attained ; e .g. However, the fact that numbers may be expressed in more than one way as the sum of two squares indicates that this bound cannot be attained for n > 15. A result of LANDAU [4] states that the number of integers less than x expressible as the sum of two squares is asymptotically c1 x (logx)-12 , so we can replace the right member of (1) by c 2 n 2 (logn)-12 and we have the upper bound but it lacks conviction since the corresponding argument in one dimension gives a false result. On the other hand we can show k > 1i2/3-E (4) for any e > 0 and sufficiently large n, by means of the following construction. Choose points successively ; when k points have been chosen, take another so that (a) it does not lie on any circle leaving one of the k points as centre and one of the Ck 2 distinct distances determined by these points as radius. (b) it does not form, with any of the first k points, a line with slope bla, (a, b) = 1, aI < n 1 / 3 I b < 0 13. Note that in particular no two points determine a distance less than n 1/3