Dot product rearrangements (Q793918)

Let \(a=(a_n)\) and \(x=(x_n)\) be sequences of non-negative integers. Let \(a.x=\sum a_nx_n.\) Letting \(x_{\pi}\) denote a permutation of the sequence \(x\), this paper investigates which subsets of \({\mathbb{R}}\) can be realised as \(a.x_{\pi}\). The main result is that if \(a_n\) increases unboundedly and \(x_n\) is positive and decreases to zero, then the set of numbers in question is the interval \([a.x,\infty]\) if and only if \(a_{n+1}/a_n\) is uniformly bounded.