On a property of functions on the sphere and its application
The author deals with a problem which is related to Knaster's conjecture. Let \(p\) be a prime, let \(q\) be even and \(m\) a natural number such that \(q\geq mp^k-mp^{k-1}\). Let \(f:S^q\to\mathbb{R}^m\) be continuous and let \(u_1,u_2,\dotsc,u_{p^k}\in S^q\) such that \(u_j\cdot u_{j+1}=\lambda\) for \(1\leq j\leq p^k\) where \(u-{p^k+1}=u_1\). The author proves that there is a rotation \(r\in\text{SO}(q+1)\) such that \[ \begin{matrix} f(ru_1)&=&f(ru_{p^{k-1}+1})&=&f(ru_{2p^{k-1}+1})&=&\dotsb&=&f(ru_{p^k-p^{k-1}+1})\\ f(ru_2)&=&f(ru_{p^{k-1}+2})&=&f(ru_{2p^{k-1}+2})&=&\dotsb&=&f(ru_{p^k-p^{k-1}+2})\\ f(ru_{p^{k-1}})&=&f(ru_{2p^{k-1}})&=&f(ru_{3p^{k-1}})&=&\dotsb&=&f(ru_{p^jk}) \end{matrix} \]
scientific article; zbMATH DE number 5800108