2018/10/27 by Zoltan Furedi, Furedi, Zoltan, Imre Z. Ruzsa +1 · 1 citation
Mathematics · #05A16 #11K65 #40A05 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A16 #msc:11K65 #msc:40A05
paper · pdf · doi:10.48550/arxiv.1810.11723
11 pages
arxiv created 2018/10/27 · arxiv updated 2018/10/30
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≤ f(1)≤ f(2)≤ f(3)≤ … such that ∑ n=1∞ f(n)/n2 = ∞. Then, there exists a sequence \b(n)\n=1,2,… satisfying b(n+m) ≤ b(n) + b(m) + f(n+m) such that the sequence of slopes \ b(n)/n\n=1,2,… takes every rational number. When the series is bounded we improve their result as follows. If there exist N and real μ>1 such that near f-subadditivity holds for all pairs (n,m) with N≤ n≤ m ≤ μn, then limn b(n)/n exists.