Dot product rearrangements

Abstract

Let a=(an), x=(xn) denote nonnegative sequences; x=(xπ(n)) denotes the rearranged sequence determined by the permutation π, a⋅x denotes the dot product ∑anxn; and S(a,x) denotes {a⋅xπ:π is a permuation of the positive integers}. We examine S(a,x) as a subset of the nonnegative real line in certain special circumstances. The main result is that if an↑∞, then S(a,x)=[a⋅x,∞] for every xn↓≠0 if and only if an+1/an is uniformly bounded.