科研之友
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摘要
An n-point k-distance set on the unit sphere S-t subset of Rt+ 1 is a set X of n points on S-t such that exactly k Euclidean distances occur between two distinct points in X. In this paper we treat distance sets on S-1, and show that if k is sufficiently small relative to n, then X lies on a regular polygon. More precisely, we prove that for an n-point k-distance set X on S1 with n >= 4, if k < 3t or 3t-2 according to whether n = 4t, 4t-1 or n = 4t-2, 4t-3, respectively, then X lies on a regular 2k or (2k + 1)-sided polygon. Furthermore, we see that this bound cannot be further improved. In addition, we find an application of Kneser's addition theorem to distance sets on circles.
- 出版日期2017-3
