Constructions of spherical 3-designs
A finite set of points \(X\) on the \(d\)-dimensional Euclidean sphere \(\mathbb S^d\subset\mathbb R^{d+1}\) is called spherical \(t\)-design if \[ \frac{1}{\sigma_d(\mathbb S^d)}\int_{\mathbb S^d} f(x)\,d\sigma_d(x)=\frac{1}{| X|}\sum_{x\in X}f(x) \] for all polynomials \(f(x)=f(x_0,x_1,\ldots,x_d)\) of total degree at most \(t\) (\(\sigma_d\) is the surface measure on \(\mathbb S^d\)). In the paper under review, new spherical 3-designs are constructed in the following cases: \(d=1\) and \(| X|\geq 4\); \(d=2\) and \(| X|=6,8\) or \(| X|\geq 10\); \(d=3\) and \(| X|=8\) or \(| X|\geq 10\); \(d=4\) and \(| X|=10,12\) or \(| X|\geq 14\); \(d=5\) and \(| X|\geq 5(d+1)/2\) odd and \(| X|\geq 2d+2\) even. On the other hand, \textit{D. Danev}, \textit{S. Nikova} and the reviewer [Discrete Comput. Geom. 21, No. 1, 143--156 (1999; Zbl 0921.05016)] have shown many nonexistence results which, in particular, add evidence for the author's conjecture that the above list of cardinalities is complete. The tools used for constructions are also interesting. Sets of integers, which generalize the classical Sidon sequences are introduced and investigated. It can be proved that Conjecture 4.2(i) is true.
scientific article; zbMATH DE number 1182269