Exact order of subsets of asymptotic bases in additive number theory (Q1098881)

Let A be an asymptotic basis of order h. Let \(I_ k(A)=\{F:\) \(F\subseteq A\), \(| F| =k\) and \(A\setminus F\) is a basis\(\}\) and g(A) the exact order of A. Define \(G_ k(h)=\max_{A;g(A)\leq h}\max_{F\in I_ k(A)}g(A\setminus F).\) The author proves that \[ G_ k(h)\geq \frac{4}{3}(\frac{h}{k+1})^{k+1}+O(h^ k) \] as h tends to infinity. The author also obtains estimates of \(G_ k(h)\) as k tends to infinity, for any fixed integer h, and proves that \(G_ k(h)\) has order of magnitude \(k^{h-1}\), \(h\geq 2\).