International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 3, Pages 409-418

Dot product rearrangements

Paul Erdos and Gary Weiss

Mathematics Institute, University of Cincinnati, Budapest, Hungary

Received 18 September 1980

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.


Let a=(an), x=(xn) denote nonnegative sequences; x=(xπ(n)) denotes the rearranged sequence determined by the permutation π, ax denotes the dot product anxn; and S(a,x) denotes {axπ: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)=[ax,] for every xn0 if and only if an+1/an is uniformly bounded.