Bifurcations in the Diophant Moment Problem

Abstract

We consider the diophant moment problem

$${n_k} = \int\limits_0^ \wedge {{\lambda ^k}d\mu (\lambda )k = 0,1,2, \ldots } $$

where dμ(λ) is a positive measure and the n_k’s are positive integers. The fact that the n_k’s are integers puts stringent constraints on the measure dμ(λ). One of the simplest results we obtain is that for Λ < 4, dμ(λ) is a finite sum of Dirac measures. An infinite number of bifurcation points in the parameter A occur in this problem. They are located at the points 4 cos² π/m, m = 2,3,4,… with an accumulation point at Λ = 4.