We show that, for a constant-degree algebraic curve γ in ℝD, every set of n points on γ spans at least Ω(n4/3) distinct distances, unless γ is an algebraic helix, in the sense of Charalambides [2]. This improves the earlier bound Ω(n5/4) of Charalambides [2].We also show that, for every set P of n points that lie on a d-dimensional constant-degree algebraic variety V in ℝD, there exists a subset S ⊂ P of size at least Ω(n4/(9+12(d−1))), such that S spans $\left({\begin{array}{*{20}{c}} {|S|} \\ 2 \\\end{array}} \right)$ distinct distances. This improves the earlier bound of Ω(n1/(3d)) of Conlon, Fox, Gasarch, Harris, Ulrich and Zbarsky [4].Both results are consequences of a common technical tool.

中文翻译：

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关于不同距离的注释

我们证明，对于恒度代数曲线γ在ℝD, 每组n点在γ跨越至少Ω（n4/3) 不同的距离，除非γ是一个代数螺旋，在 Charalambides [2] 的意义上。这改进了早先的界限 Ω(n5/4) 的 Charalambides [2]。我们还表明，对于每个集合磷的n位于 a 上的点d维常度代数簇五在ℝD, 存在一个子集小号⊂磷尺寸至少 Ω(n4/(9+12(d-1)))，这样小号跨度$\left({\begin{array}{*{20}{c}} {|S|} \\ 2 \\\end{array}} \right)$不同的距离。这改进了 Ω(n1/(3d)) 的 Conlon、Fox、Gasarch、Harris、Ulrich 和 Zbarsky [4]。这两个结果都是通用技术工具的结果。