AI 中文总结
该研究在任意维数下构造了仿射一般位置的无限有理距离集，奇数维时还可实现无 $d+2$ 点共球，三维时得到无限多非相似本原 $n_3$-簇，为相关几何问题提供了统一构造方法。
AI 中文摘要
对于每个整数 $d\geq 1$，我们构造了一个位于 $\mathbb{R}^d$ 中的可数无限集 $X_d$，它处于仿射一般位置，且所有两两距离均为有理数。当 $d$ 为奇数时，$X_d$ 还可被构造为不存在 $d+2$ 个点共球。该构造在 $d$ 上具有一致性：正切比雪夫方块分解在球面上生成调和曲线，对应有理参数值的点具有两两有理距离。通过仿射行列式的分差因式分解可知，足够短的弧是局部凸的，而球极投影可生成奇数维的例子。我们还对每个偶数 $d$ 构造了 $\mathbb{Q}^d$ 中处于仿射一般位置的无限有理距离集，对每个 $d\equiv 1\pmod 4$ 构造了处于一般位置的无限有理距离集。对于每个 $d\geq 1$ 和 $n\geq d+1$，通过选取并缩放合适的有限子集，可得到处于仿射一般位置的 $n$ 点整数点集；合适的有序选择可生成所有循环多面体的整数距离实现。在三维空间中，我们给出了显式有理参数化，并对每个 $n\geq 4$ 得到了无限多个两两非相似的本原 $n_3$-簇。
英文摘要
For every integer $d\geq 1$, we construct a countably infinite set $X_d\subset\mathbb{R}^d$ in affine general position, with all pairwise distances rational. When $d$ is odd, $X_d$ may also be chosen so that no $d+2$ points lie on a common sphere. The construction is uniform in $d$: positive Chebyshev square decompositions produce harmonic curves on spheres whose points corresponding to rational parameter values have pairwise rational distances. A divided-difference factorization of the affine determinant shows that sufficiently short arcs are locally convex, and stereographic projection produces the odd-dimensional examples. We also construct infinite rational distance sets in $\mathbb{Q}^d$ in affine general position for every even $d$, and in general position for every $d\equiv 1\pmod 4$. For every $d\geq 1$ and $n\geq d+1$, taking and rescaling suitable finite subsets gives $n$-point integral point sets in affine general position. A suitable ordered choice yields integral-distance realizations of all cyclic polytopes. In dimension three, we give an explicit rational parametrization and obtain infinitely many pairwise non-similar primitive $n_3$-clusters for every $n\geq 4$.
Comments27 pages; a 7-page computational supplement and exact verification code are included as ancillary files
