Sets with a Prescribed Number of Power Invariants

Let A₁,...,A_n be points in , let be a fixed point, let p be a positive integer, and let ₁,...,_n be positive real numbers. If the does not depend on the position of M on a sphere with center O, then one says that the point system {A₁,...,A_n} has an invariant of degree p with weight system {,...,_n}. It is proved that for arbitrary positive integers d and N there exists a finite point system having invariants of degrees p=1,...,N with common positive weight system {₁,...,_n}. Bibliography: 2 titles.