Dot product rearrangements

Let a = ( a n ) , x = ( x n ) denote nonnegative sequences; x = ( x π ( n ) ) denotes the rearranged sequence determined by the permutation π , a ⋅ x denotes the dot product ∑ a n x n ; and S ( a , x ) denotes { a ⋅ x π : 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 a n ↑ ∞ , then S ( a , x ) = [ a ⋅ x , ∞ ] for every x n ↓ ≠ 0 if and only if a n + 1 / a n is uniformly bounded.