Paul Erdos and Gary Weiss

Copyright © 1983 Paul Erdos and Gary Weiss. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.