A basis of order h for the integers is a set of integers such that the h-fold sum set, A+A+A...+A (h times) is the set of all nonnegative integers. The smallest possible bases of order h have |A \cap {1,2,3,...,n}| \sim c n^{1/h} for some constant c>0. This fact led to a study of strictly increasing sequences of integers asymptotic to various growth functions.
I explored the following question: Let f(n) be a growth function, and A be a sequence with f(n) \leq a_n \leq Uf(n), U constant. Under what conditions is it possible to construct a sequence B, with b_k \sim \beta f(k), which has A as a subsequence?
Brooke Orosz