Nearly subadditive sequences

Abstract

We show that the de Bruijn-Erdős condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. Suppose that given a sequence 0(1)Then, there exists a sequence {b(nsuch that the sequence of slopes {b(n)/n} n=1,2,... takes every rational number.When the series ( 1) is bounded we improve their result as follows. If there exist N and real µ > 1 such that (2) holds for all pairs (n, m) with N ≤ n ≤ m ≤ µn, then lim n b(n)/n exists.